#set document(title: "9.4 Use Properties of Rectangles, Triangles, and Trapezoids", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.4#h(0.6em)Use Properties of Rectangles, Triangles, and Trapezoids #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. The length of a rectangle is #math.equation(block: false, alt: "3")[$3$] less than the width. Let #math.equation(block: false, alt: "w")[$w$] represent the width. Write an expression for the length of the rectangle. #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "w minus 3")[$w − 3$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 1 over 2 open parenthesis 6 h close parenthesis .")[$frac(1, 2) ( 6 h ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "3 h")[$3 h$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the fraction 5 over 2 open parenthesis 10.3 minus 7.9 close parenthesis .")[$frac(5, 2) ( 10.3 − 7.9 ) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "6")[$6$] ] ] In this section, we’ll continue working with geometry applications. We will add some more properties of triangles, and we’ll learn about the properties of rectangles and trapezoids. === Understand Linear, Square, and Cubic Measure When you measure your height or the length of a garden hose, you use a ruler or tape measure . A tape measure might remind you of a line—you use it for #strong[linear measure], which measures length. Inch, foot, yard, mile, centimeter and meter are units of linear measure. #figure(figph[A picture of a portion of a tape measure is shown. The top shows the numbers 1 through 5. The portion from the beginning to the 1 has a red circle and an arrow to a picture from 0 to 1 inch, with 1 sixteenth, 1 eighth, 3 eighths, 1 half, and 3 fourths labeled. Above this, it is labeled “Standard Measures.” The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2. The region from the edge to about 3 and a half has a red circle with an arrow pointing to a picture from 0 to 3.5. It is labeled 0, 1 cm, 1.7 cm, 2.3 cm and 3.5 cm. Above this, it is labeled “Metric (S).”], alt: "A picture of a portion of a tape measure is shown. The top shows the numbers 1 through 5. The portion from the beginning to the 1 has a red circle and an arrow to a picture from 0 to 1 inch, with 1 sixteenth, 1 eighth, 3 eighths, 1 half, and 3 fourths labeled. Above this, it is labeled “Standard Measures.” The bottom of the tape measure shows the numbers 1 through 10, then 1 and 2. The region from the edge to about 3 and a half has a red circle with an arrow pointing to a picture from 0 to 3.5. It is labeled 0, 1 cm, 1.7 cm, 2.3 cm and 3.5 cm. Above this, it is labeled “Metric (S).”", caption: [This tape measure measures inches along the top and centimeters along the bottom.]) When you want to know how much tile is needed to cover a floor, or the size of a wall to be painted, you need to know the #strong[area], a measure of the region needed to cover a surface. Area is measured is #strong[square units]. We often use square inches, square feet, square centimeters, or square miles to measure area. A square centimeter is a square that is one centimeter (cm) on each side. A square inch is a square that is one inch on each side . #figure(figph[Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimeter. The larger one has sides labeled 1 inch and is 1 square inch.], alt: "Two squares are shown. The smaller one has sides labeled 1 cm and is 1 square centimeter. The larger one has sides labeled 1 inch and is 1 square inch.", caption: [Square measures have sides that are each #math.equation(block: false, alt: "1")[$1$] unit in length.]) shows a rectangular rug that is #math.equation(block: false, alt: "2")[$2$] feet long by #math.equation(block: false, alt: "3")[$3$] feet wide. Each square is #math.equation(block: false, alt: "1")[$1$] foot wide by #math.equation(block: false, alt: "1")[$1$] foot long, or #math.equation(block: false, alt: "1")[$1$] square foot. The rug is made of #math.equation(block: false, alt: "6")[$6$] squares. The area of the rug is #math.equation(block: false, alt: "6")[$6$] square feet. #figure(figph[A rectangle is shown. It has 3 squares across and 2 squares down, a total of 6 squares.], alt: "A rectangle is shown. It has 3 squares across and 2 squares down, a total of 6 squares.", caption: [The rug contains six squares of 1 square foot each, so the total area of the rug is 6 square feet.]) When you measure how much it takes to fill a container, such as the amount of gasoline that can fit in a tank, or the amount of medicine in a syringe, you are measuring #strong[volume]. Volume is measured in #strong[cubic units] such as cubic inches or cubic centimeters. When measuring the volume of a rectangular solid, you measure how many cubes fill the container. We often use cubic centimeters, cubic inches, and cubic feet. A cubic centimeter is a cube that measures one centimeter on each side, while a cubic inch is a cube that measures one inch on each side . #figure(figph[Two cubes are shown. The smaller one has sides labeled 1 cm and is labeled as 1 cubic centimeter. The larger one has sides labeled 1 inch and is labeled as 1 cubic inch.], alt: "Two cubes are shown. The smaller one has sides labeled 1 cm and is labeled as 1 cubic centimeter. The larger one has sides labeled 1 inch and is labeled as 1 cubic inch.", caption: [Cubic measures have sides that are 1 unit in length.]) Suppose the cube in measures #math.equation(block: false, alt: "3")[$3$] inches on each side and is cut on the lines shown. How many little cubes does it contain? If we were to take the big cube apart, we would find #math.equation(block: false, alt: "27")[$27$] little cubes, with each one measuring one inch on all sides. So each little cube has a volume of #math.equation(block: false, alt: "1")[$1$] cubic inch, and the volume of the big cube is #math.equation(block: false, alt: "27")[$27$] cubic inches. #figure(figph[A cube is shown, comprised of smaller cubes. Each side of the cube has 3 smaller cubes across, for a total of 27 smaller cubes.], alt: "A cube is shown, comprised of smaller cubes. Each side of the cube has 3 smaller cubes across, for a total of 27 smaller cubes.", caption: [A cube that measures 3 inches on each side is made up of 27 one-inch cubes, or 27 cubic inches.]) #examplebox("Example 1")[][ For each item, state whether you would use linear, square, or cubic measure: + ⓐ amount of carpeting needed in a room + ⓑ extension cord length + ⓒ amount of sand in a sandbox + ⓓ length of a curtain rod + ⓔ amount of flour in a canister + ⓕ size of the roof of a doghouse. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ You are measuring how much surface the carpet covers, which is the area.], [square measure]), [ⓑ You are measuring how long the extension cord is, which is the length.], [linear measure], [ⓒ You are measuring the volume of the sand.], [cubic measure], [ⓓ You are measuring the length of the curtain rod.], [linear measure], [ⓔ You are measuring the volume of the flour.], [cubic measure], [ⓕ You are measuring the area of the roof.], [square measure], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether you would use linear, square, or cubic measure for each item. ⓐ amount of paint in a can ⓑ height of a tree ⓒ floor of your bedroom ⓓ diameter of bike wheel ⓔ size of a piece of sod ⓕ amount of water in a swimming pool #solutionbox[ + ⓐ cubic + ⓑ linear + ⓒ square + ⓓ linear + ⓔ square + ⓕ cubic ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Determine whether you would use linear, square, or cubic measure for each item. ⓐ volume of a packing box ⓑ size of patio ⓒ amount of medicine in a syringe ⓓ length of a piece of yarn ⓔ size of housing lot ⓕ height of a flagpole #solutionbox[ + ⓐ cubic + ⓑ square + ⓒ cubic + ⓓ linear + ⓔ square + ⓕ linear ] ] Many geometry applications will involve finding the perimeter or the area of a figure. There are also many applications of perimeter and area in everyday life, so it is important to make sure you understand what they each mean. Picture a room that needs new floor tiles. The tiles come in squares that are a foot on each side—one square foot. How many of those squares are needed to cover the floor? This is the area of the floor. Next, think about putting new baseboard around the room, once the tiles have been laid. To figure out how many strips are needed, you must know the distance around the room. You would use a tape measure to measure the number of feet around the room. This distance is the perimeter. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Perimeter and Area] The #strong[perimeter] is a measure of the distance around a figure. The #strong[area] is a measure of the surface covered by a figure. ] shows a square tile that is #math.equation(block: false, alt: "1")[$1$] inch on each side. If an ant walked around the edge of the tile, it would walk #math.equation(block: false, alt: "4")[$4$] inches. This distance is the perimeter of the tile. Since the tile is a square that is #math.equation(block: false, alt: "1")[$1$] inch on each side, its area is one square inch. The area of a shape is measured by determining how many square units cover the shape. #figure(figph[A 5 square by 5 square checkerboard is shown with each side labeled 1 inch. An image of an ant is shown on the top left square.], alt: "A 5 square by 5 square checkerboard is shown with each side labeled 1 inch. An image of an ant is shown on the top left square.", caption: [#math.equation(block: false, alt: "Perimeter equals 4 inches; Area equals 1 square inch")[$"Perimeter" = 4 #h(0.2em) "inches" \ "Area" = 1 #h(0.2em) "square inch"$] #linebreak() When the ant walks completely around the tile on its edge, it is tracing the perimeter of the tile. The area of the tile is 1 square inch.]) #examplebox("Example 2")[][ Each of two square tiles is #math.equation(block: false, alt: "1")[$1$] square inch. Two tiles are shown together. + ⓐ What is the perimeter of the figure? + ⓑ What is the area? #figure(figph[A checkerboard is shown. It has 10 squares across the top and 5 down the side.], alt: "A checkerboard is shown. It has 10 squares across the top and 5 down the side.", caption: none) #solutionbox[ ⓐ The perimeter is the distance around the figure. The perimeter is #math.equation(block: false, alt: "6")[$6$] inches. ⓑ The area is the surface covered by the figure. There are #math.equation(block: false, alt: "2")[$2$] square inch tiles so the area is #math.equation(block: false, alt: "2")[$2$] square inches. #figure(figph[A checkerboard is shown. It has 10 squares across the top and 5 down the side. The top and bottom each have two adjacent 1 inch labels across, the sides have 1 inch labels.], alt: "A checkerboard is shown. It has 10 squares across the top and 5 down the side. The top and bottom each have two adjacent 1 inch labels across, the sides have 1 inch labels.", caption: none) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure: #figure(figph[A rectangle is shown comprised of 3 squares.], alt: "A rectangle is shown comprised of 3 squares.", caption: none) #solutionbox[ + ⓐ 8 inches + ⓑ 3 sq. inches ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Each box in the figure below is 1 square inch. Find the ⓐ perimeter and ⓑ area of the figure: #figure(figph[A square is shown comprised of 4 smaller squares.], alt: "A square is shown comprised of 4 smaller squares.", caption: none) #solutionbox[ + ⓐ 8 centimeters + ⓑ 4 sq. centimeters ] ] === Use the Properties of Rectangles A #strong[rectangle] has four sides and four right angles. The opposite sides of a rectangle are the same length. We refer to one side of the rectangle as the length, #math.equation(block: false, alt: "L ,")[$L ,$] and the adjacent side as the width, #math.equation(block: false, alt: "W .")[$W .$] #figure(figph[A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.], alt: "A rectangle is shown. Each angle is marked with a square. The top and bottom are labeled L, the sides are labeled W.", caption: [A rectangle has four sides, and four right angles. The sides are labeled L for length and W for width.]) The perimeter, #math.equation(block: false, alt: "P ,")[$P ,$] of the rectangle is the distance around the rectangle. If you started at one corner and walked around the rectangle, you would walk #math.equation(block: false, alt: "L plus W plus L plus W")[$L + W + L + W$] units, or two lengths and two widths. The perimeter then is #math.equation(block: true, alt: "P equals L plus W plus L plus W; or; P equals 2 L plus 2 W")[$P = L + W + L + W \ "or" \ P = 2 L + 2 W$]What about the area of a rectangle? Remember the rectangular rug from the beginning of this section. It was #math.equation(block: false, alt: "2")[$2$] feet long by #math.equation(block: false, alt: "3")[$3$] feet wide, and its area was #math.equation(block: false, alt: "6")[$6$] square feet. Since #math.equation(block: false, alt: "A equals 2 times 3 ,")[$A = 2 ⋅ 3 ,$] we see that the area, #math.equation(block: false, alt: "A ,")[$A ,$] is the length, #math.equation(block: false, alt: "L ,")[$L ,$] times the width, #math.equation(block: false, alt: "W ,")[$W ,$] so the area of a rectangle is #math.equation(block: false, alt: "A equals L times W .")[$A = L ⋅ W .$] #figure(figph[A rectangle is shown. It is made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.], alt: "A rectangle is shown. It is made up of 6 squares. The bottom is 2 squares across and marked as 2, the side is 3 squares long and marked as 3.", caption: [The area of this rectangular rug is #math.equation(block: false, alt: "6")[$6$] square feet, its length times its width.]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Rectangles] - Rectangles have four sides and four right #math.equation(block: false, alt: "open parenthesis 90° close parenthesis")[$( "90°" )$] angles. - The lengths of opposite sides are equal. - The perimeter, #math.equation(block: false, alt: "P ,")[$P ,$] of a rectangle is the sum of twice the length and twice the width. #linebreak() #math.equation(block: true, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$] - The area, #math.equation(block: false, alt: "A ,")[$A ,$] of a rectangle is the length times the width. #linebreak() #math.equation(block: true, alt: "A equals L times W")[$A = L ⋅ W$] ] For easy reference as we work the examples in this section, we will restate the Problem Solving Strategy for Geometry Applications here. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Use a Problem Solving Strategy for Geometry Applications] + #strong[Read] the problem and make sure you understand all the words and ideas. Draw the figure and label it with the given information. + #strong[Identify] what you are looking for. + #strong[Name] what you are looking for. Choose a variable to represent that quantity. + #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. + #strong[Solve] the equation using good algebra techniques. + #strong[Check] the answer in the problem and make sure it makes sense. + #strong[Answer] the question with a complete sentence. ] #examplebox("Example 3")[][ The length of a rectangle is #math.equation(block: false, alt: "32")[$32$] meters and the width is #math.equation(block: false, alt: "20")[$20$] meters. Find ⓐ the perimeter, and ⓑ the area. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A rectangle is shown with sides labeled 32 m and 20 m, indicating its length and width.], alt: "A rectangle is shown with sides labeled 32 m and 20 m, indicating its length and width.", caption: none)], [Step 2. #strong[Identify] what you are looking for.], [the perimeter of a rectangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[P] = the perimeter], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[The formula for the perimeter of a rectangle, P = 2L + 2W, with L=32 and W=20 substituted as P = 2(32) + 2(20).], alt: "The formula for the perimeter of a rectangle, P = 2L + 2W, with L=32 and W=20 substituted as P = 2(32) + 2(20).", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation showing P = 64 + 40, which is then solved to P = 104.], alt: "A mathematical equation showing P = 64 + 40, which is then solved to P = 104.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical equation solving for P, showing the sum of 20 + 32 + 20 + 32 being verified as equal to 104. The final line confirms '104 = 104' with a checkmark.], alt: "A mathematical equation solving for P, showing the sum of 20 + 32 + 20 + 32 being verified as equal to 104. The final line confirms '104 = 104' with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The perimeter of the rectangle is 104 meters.], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A rectangle with a length of 32 meters and a width of 20 meters, clearly indicating its dimensions.], alt: "A rectangle with a length of 32 meters and a width of 20 meters, clearly indicating its dimensions.", caption: none)], [Step 2. #strong[Identify] what you are looking for.], [the area of a rectangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[A] = the area], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[The image displays the formula for the area of a rectangle, A = L x W, with an example showing A = 32 m x 20 m. Brackets underneath each variable and operator indicate what each symbol represents.], alt: "The image displays the formula for the area of a rectangle, A = L x W, with an example showing A = 32 m x 20 m. Brackets underneath each variable and operator indicate what each symbol represents.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[The image displays the equation 'A = 640' centered on a plain white background, rendered in a standard, dark font. It appears to be a mathematical or numerical representation.], alt: "The image displays the equation 'A = 640' centered on a plain white background, rendered in a standard, dark font. It appears to be a mathematical or numerical representation.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical verification showing A equals 32 multiplied by 20, which simplifies to 640, thereby confirming the equation 640 = 640 with a checkmark.], alt: "A mathematical verification showing A equals 32 multiplied by 20, which simplifies to 640, thereby confirming the equation 640 = 640 with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The area of the rectangle is 640 square meters.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is #math.equation(block: false, alt: "120")[$120$] yards and the width is #math.equation(block: false, alt: "50")[$50$] yards. Find ⓐ the perimeter and ⓑ the area. #solutionbox[ + ⓐ 340 yd + ⓑ 6000 sq. yd ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is #math.equation(block: false, alt: "62")[$62$] feet and the width is #math.equation(block: false, alt: "48")[$48$] feet. Find ⓐ the perimeter and ⓑ the area. #solutionbox[ + ⓐ 220 ft + ⓑ 2976 sq. ft ] ] #examplebox("Example 4")[][ Find the length of a rectangle with perimeter #math.equation(block: false, alt: "50")[$50$] inches and width #math.equation(block: false, alt: "10")[$10$] inches. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A rectangle is depicted, having a height of 10 inches on both vertical sides and a width labeled 'L' on both horizontal sides, indicating its dimensions.], alt: "A rectangle is depicted, having a height of 10 inches on both vertical sides and a width labeled 'L' on both horizontal sides, indicating its dimensions.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the length of the rectangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[L] = the length], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Mathematical equations showing the perimeter formula P = 2L + 2W. Below it, P is substituted with 50 and W with 10, leading to 50 = 2L + 2(10).], alt: "Mathematical equations showing the perimeter formula P = 2L + 2W. Below it, P is substituted with 50 and W with 10, leading to 50 = 2L + 2(10).", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[Mathematical steps showing the solution of a linear equation. The equation 50 - 20 = 2L + 20 - 20 is simplified to 30 = 2L, and then L is found to be 15 by dividing by 2.], alt: "Mathematical steps showing the solution of a linear equation. The equation 50 - 20 = 2L + 20 - 20 is simplified to 30 = 2L, and then L is found to be 15 by dividing by 2.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical equation checks if 15 + 10 + 15 + 10 equals 50, confirming the sum is indeed 50 with a checkmark, potentially demonstrating a perimeter calculation.], alt: "A mathematical equation checks if 15 + 10 + 15 + 10 equals 50, confirming the sum is indeed 50 with a checkmark, potentially demonstrating a perimeter calculation.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The length is 15 inches.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the length of a rectangle with a perimeter of #math.equation(block: false, alt: "80")[$80$] inches and width of #math.equation(block: false, alt: "25")[$25$] inches. #solutionbox[ 15 in. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the length of a rectangle with a perimeter of #math.equation(block: false, alt: "30")[$30$] yards and width of #math.equation(block: false, alt: "6")[$6$] yards. #solutionbox[ 9 yd ] ] In the next example, the width is defined in terms of the length. We’ll wait to draw the figure until we write an expression for the width so that we can label one side with that expression. #examplebox("Example 5")[][ The width of a rectangle is two inches less than the length. The perimeter is #math.equation(block: false, alt: "52")[$52$] inches. Find the length and width. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [the length and width of the rectangle], [Step 3. #strong[Name.] Choose a variable to represent it. #linebreak() #linebreak() Now we can draw a figure using these expressions for the length and width.], [Since the width is defined in terms of the length, we let #emph[L] = length. The width is two feet less that the length, so we let #emph[L] − 2 = width #linebreak() #figure(figph[A square with its top and bottom sides labeled 'L' and its left and right sides labeled 'L - 2'.], alt: "A square with its top and bottom sides labeled 'L' and its left and right sides labeled 'L - 2'.", caption: none)], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. The formula for the perimeter of a rectangle relates all the information. #linebreak() Substitute in the given information.], [#linebreak() #figure(figph[A mathematical equation illustrating the perimeter of a rectangle, where P = 2L + 2W is shown, followed by the specific values and substitution 52 = 2L + 2(L-2).], alt: "A mathematical equation illustrating the perimeter of a rectangle, where P = 2L + 2W is shown, followed by the specific values and substitution 52 = 2L + 2(L-2).", caption: none)], [Step 5. #strong[Solve] the equation.], [#math.equation(block: false, alt: "52 equals 2 L plus 2 L minus 4")[$52 = 2 L + 2 L − 4$]], [Combine like terms.], [#math.equation(block: false, alt: "52 equals 4 L − 4")[$52 = 4 L − 4$]], [Add 4 to each side.], [#math.equation(block: false, alt: "56 equals 4 L")[$56 = 4 L$]], [Divide by 4.], [#math.equation(block: false, alt: "the fraction 56 over 4 equals the fraction 4 L over 4")[$frac(56, 4) = frac(4 L, 4)$]], [], [#math.equation(block: false, alt: "14 equals L")[$14 = L$]], [], [The length is 14 inches.], [Now we need to find the width.], [], [The width is #emph[L] − 2.], [#figure(figph[An algebraic substitution showing L-2, then substituting L with 14 (highlighted in red) to get 14-2, which equals 12.], alt: "An algebraic substitution showing L-2, then substituting L with 14 (highlighted in red) to get 14-2, which equals 12.", caption: none) #linebreak() The width is 12 inches.], [Step 6. #strong[Check:] #linebreak() Since #math.equation(block: false, alt: "14 plus 12 plus 14 plus 12 equals 52")[$14 + 12 + 14 + 12 = 52$], this works!], [], [Step 7. #strong[Answer] the question.], [The length is 14 feet and the width is 12 feet.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The width of a rectangle is seven meters less than the length. The perimeter is #math.equation(block: false, alt: "58")[$58$] meters. Find the length and width. #solutionbox[ 18 m, 11 m ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is eight feet more than the width. The perimeter is #math.equation(block: false, alt: "60")[$60$] feet. Find the length and width. #solutionbox[ 11 ft , 19 ft ] ] #examplebox("Example 6")[][ The length of a rectangle is four centimeters more than twice the width. The perimeter is #math.equation(block: false, alt: "32")[$32$] centimeters. Find the length and width. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [the length and width], [Step 3. #strong[Name.] Choose a variable to represent it.], [let #emph[W] = width #linebreak() The length is four more than twice the width. #linebreak() 2#emph[w] + 4 = length #linebreak() #figure(figph[A rectangle is depicted with its dimensions labeled algebraically. The width is denoted by 'w' and the length is denoted by '2w + 4'.], alt: "A rectangle is depicted with its dimensions labeled algebraically. The width is denoted by 'w' and the length is denoted by '2w + 4'.", caption: none)], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula and substitute in the given information.], [#figure(figph[The image displays a two-line mathematical equation. The first line presents the formula for the perimeter of a rectangle, P = 2L + 2W. The second line shows a specific application of this formula: 32 = 2(2w + 4) + 2w.], alt: "The image displays a two-line mathematical equation. The first line presents the formula for the perimeter of a rectangle, P = 2L + 2W. The second line shows a specific application of this formula: 32 = 2(2w + 4) + 2w.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[An algebraic solution for 'w' from the equation 32 = 4w + 8 + 2w, showing w = 4 (width). The length, calculated as 2w + 4, is found to be 12 cm.], alt: "An algebraic solution for 'w' from the equation 32 = 4w + 8 + 2w, showing w = 4 (width). The length, calculated as 2w + 4, is found to be 12 cm.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A step-by-step mathematical verification of the perimeter formula p = 2L + 2W, using values L=12 and W=4 to show 32 = 32.], alt: "A step-by-step mathematical verification of the perimeter formula p = 2L + 2W, using values L=12 and W=4 to show 32 = 32.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The length is 12 cm and the width is 4 cm.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangle is eight more than twice the width. The perimeter is #math.equation(block: false, alt: "64")[$64$] feet. Find the length and width. #solutionbox[ 8 ft, 24 ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The width of a rectangle is six less than twice the length. The perimeter is #math.equation(block: false, alt: "18")[$18$] centimeters. Find the length and width. #solutionbox[ 5 cm, 4 cm ] ] #examplebox("Example 7")[][ The area of a rectangular room is #math.equation(block: false, alt: "168")[$168$] square feet. The length is #math.equation(block: false, alt: "14")[$14$] feet. What is the width? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], [#figure(figph[A rectangle is shown with width 'W', length '14 ft', and an area of '168 ft^2'.], alt: "A rectangle is shown with width 'W', length '14 ft', and an area of '168 ft^2'.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the width of a rectangular room], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[W] = width], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula and substitute in the given information.], [#figure(figph[Two equations are displayed: A = LW (Area = Length x Width) and 168 = 14W. This represents a geometry problem where the area is 168 and the length is 14, seeking the width W.], alt: "Two equations are displayed: A = LW (Area = Length x Width) and 168 = 14W. This represents a geometry problem where the area is 168 and the length is 14, seeking the width W.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation shows 168 divided by 14 equals 14W divided by 14. Below this, the solution is shown as 12 equals W, indicating the variable W has been solved for.], alt: "A mathematical equation shows 168 divided by 14 equals 14W divided by 14. Below this, the solution is shown as 12 equals W, indicating the variable W has been solved for.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical verification is shown, starting with the area formula A=LW, then questioning if 168 equals 14 multiplied by 12, and finally confirming that 168 does indeed equal 168.], alt: "A mathematical verification is shown, starting with the area formula A=LW, then questioning if 168 equals 14 multiplied by 12, and finally confirming that 168 does indeed equal 168.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The width of the room is 12 feet.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The area of a rectangle is #math.equation(block: false, alt: "598")[$598$] square feet. The length is #math.equation(block: false, alt: "23")[$23$] feet. What is the width? #solutionbox[ 26 ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The width of a rectangle is #math.equation(block: false, alt: "21")[$21$] meters. The area is #math.equation(block: false, alt: "609")[$609$] square meters. What is the length? #solutionbox[ 29 m ] ] #examplebox("Example 8")[][ The perimeter of a rectangular swimming pool is #math.equation(block: false, alt: "150")[$150$] feet. The length is #math.equation(block: false, alt: "15")[$15$] feet more than the width. Find the length and width. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A rectangular body of water with width W and length W+15. The perimeter of the rectangle is given as P=150 ft.], alt: "A rectangular body of water with width W and length W+15. The perimeter of the rectangle is given as P=150 ft.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the length and width of the pool], [Step 3. #strong[Name.] Choose a variable to represent it. #linebreak() The length is 15 feet more than the width.], [Let #math.equation(block: false, alt: "W equals width")[$W = "width"$] #linebreak() #math.equation(block: false, alt: "W plus 15 equals length")[$W + 15 = "length"$]], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula and substitute.], [#figure(figph[Perimeter formula P = 2L + 2W applied to a problem, substituting P=150, L=(w+15), and W=w to form the equation 150 = 2(w + 15) + 2w.], alt: "Perimeter formula P = 2L + 2W applied to a problem, substituting P=150, L=(w+15), and W=w to form the equation 150 = 2(w + 15) + 2w.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[This image demonstrates the algebraic process of finding the dimensions of a pool. It solves for 'w', the width, which is 30, and then calculates the length as 'w + 15', resulting in 45.], alt: "This image demonstrates the algebraic process of finding the dimensions of a pool. It solves for 'w', the width, which is 30, and then calculates the length as 'w + 15', resulting in 45.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[Mathematical steps demonstrating the calculation of a rectangle's perimeter. It shows the formula p = 2L + 2W, then substitutes L=45 and W=30 to verify if p=150, concluding with 150 = 150.], alt: "Mathematical steps demonstrating the calculation of a rectangle's perimeter. It shows the formula p = 2L + 2W, then substitutes L=45 and W=30 to verify if p=150, concluding with 150 = 150.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The length of the pool is 45 feet and the width is 30 feet.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The perimeter of a rectangular swimming pool is #math.equation(block: false, alt: "200")[$200$] feet. The length is #math.equation(block: false, alt: "40")[$40$] feet more than the width. Find the length and width. #solutionbox[ 30 ft, 70 ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The length of a rectangular garden is #math.equation(block: false, alt: "30")[$30$] yards more than the width. The perimeter is #math.equation(block: false, alt: "300")[$300$] yards. Find the length and width. #solutionbox[ 60 yd, 90 yd ] ] === Use the Properties of Triangles We now know how to find the area of a rectangle. We can use this fact to help us visualize the formula for the area of a triangle. In the rectangle in , we’ve labeled the length #math.equation(block: false, alt: "b")[$b$] and the width #math.equation(block: false, alt: "h ,")[$h ,$] so it’s area is #math.equation(block: false, alt: "b h .")[$b h .$] #figure(figph[A rectangle is shown. The side is labeled h and the bottom is labeled b. The center says A equals bh.], alt: "A rectangle is shown. The side is labeled h and the bottom is labeled b. The center says A equals bh.", caption: [The area of a rectangle is the base, #math.equation(block: false, alt: "b ,")[$b ,$] times the height, #math.equation(block: false, alt: "h .")[$h .$]]) We can divide this rectangle into two #strong[congruent] triangles . Triangles that are congruent have identical side lengths and angles, and so their areas are equal. The area of each triangle is one-half the area of the rectangle, or #math.equation(block: false, alt: "the fraction 1 over 2 b h .")[$frac(1, 2) b h .$] This example helps us see why the formula for the area of a triangle is #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h .")[$A = frac(1, 2) b h .$] #figure(figph[A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says “Area of each triangle,” and shows the equation A equals one-half bh.], alt: "A rectangle is shown. A diagonal line is drawn from the upper left corner to the bottom right corner. The side of the rectangle is labeled h and the bottom is labeled b. Each triangle says one-half bh. To the right of the rectangle, it says “Area of each triangle,” and shows the equation A equals one-half bh.", caption: [A rectangle can be divided into two triangles of equal area. The area of each triangle is one-half the area of the rectangle.]) The formula for the area of a triangle is #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h ,")[$A = frac(1, 2) b h ,$] where #math.equation(block: false, alt: "b")[$b$] is the base and #math.equation(block: false, alt: "h")[$h$] is the height. To find the area of the triangle, you need to know its base and height. The base is the length of one side of the triangle, usually the side at the bottom. The height is the length of the line that connects the base to the opposite vertex, and makes a #math.equation(block: false, alt: "90°")[$"90°"$] angle with the base. shows three triangles with the base and height of each marked. #figure(figph[Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.], alt: "Three triangles are shown. The triangle on the left is a right triangle. The bottom is labeled b and the side is labeled h. The middle triangle is an acute triangle. The bottom is labeled b. There is a dotted line from the top vertex to the base of the triangle, forming a right angle with the base. That line is labeled h. The triangle on the right is an obtuse triangle. The bottom of the triangle is labeled b. The base has a dotted line extended out and forms a right angle with a dotted line to the top of the triangle. The vertical line is labeled h.", caption: [The height #math.equation(block: false, alt: "h")[$h$] of a triangle is the length of a line segment that connects the the base to the opposite vertex and makes a #math.equation(block: false, alt: "90°")[$"90°"$] angle with the base.]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Triangle Properties] For any triangle #math.equation(block: false, alt: "Δ A B C ,")[$"Δ" A B C ,$] the sum of the measures of the angles is #math.equation(block: false, alt: "180° .")[$"180°" .$] #math.equation(block: true, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180°")[$m "∠" A + m "∠" B + m "∠" C = "180°"$]The perimeter of a triangle is the sum of the lengths of the sides. #math.equation(block: true, alt: "P equals a plus b plus c")[$P = a + b + c$]The area of a triangle is one-half the base, #math.equation(block: false, alt: "b ,")[$b ,$] times the height, #math.equation(block: false, alt: "h .")[$h .$] #math.equation(block: true, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) #h(0.1em) b h$]#figure(figph[A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.], alt: "A triangle is shown. The vertices are labeled A, B, and C. The sides are labeled a, b, and c. There is a vertical dotted line from vertex B at the top of the triangle to the base of the triangle, meeting the base at a right angle. The dotted line is labeled h.", caption: none) ] #examplebox("Example 9")[][ Find the area of a triangle whose base is #math.equation(block: false, alt: "11")[$11$] inches and whose height is #math.equation(block: false, alt: "8")[$8$] inches. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[An image of a triangle showing its base and height. The base is labeled as 11 inches, and the height, represented by a dashed line perpendicular to the base, is labeled as 8 inches.], alt: "An image of a triangle showing its base and height. The base is labeled as 11 inches, and the height, represented by a dashed line perpendicular to the base, is labeled as 8 inches.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the area of the triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [let #emph[A] = area of the triangle], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Two lines of an area formula: the first line shows A = 1/2 \* b \* h, and the second line substitutes b = 11 and h = 8 into the formula, showing A = 1/2 \* 11 \* 8.], alt: "Two lines of an area formula: the first line shows A = 1/2 * b * h, and the second line substitutes b = 11 and h = 8 into the formula, showing A = 1/2 * 11 * 8.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[The image displays the equation A = 44 square inches, indicating an area measurement.], alt: "The image displays the equation A = 44 square inches, indicating an area measurement.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical calculation verifying the area of a triangle using the formula A = 1/2 bh. The steps show 44 compared to 1/2(11)8, confirming the equality 44 = 44 with a checkmark.], alt: "A mathematical calculation verifying the area of a triangle using the formula A = 1/2 bh. The steps show 44 compared to 1/2(11)8, confirming the equality 44 = 44 with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The area is 44 square inches.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the area of a triangle with base #math.equation(block: false, alt: "13")[$13$] inches and height #math.equation(block: false, alt: "2")[$2$] inches. #solutionbox[ 13 sq. in. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the area of a triangle with base #math.equation(block: false, alt: "14")[$14$] inches and height #math.equation(block: false, alt: "7")[$7$] inches. #solutionbox[ 49 sq. in. ] ] #examplebox("Example 10")[][ The perimeter of a triangular garden is #math.equation(block: false, alt: "24")[$24$] feet. The lengths of two sides are #math.equation(block: false, alt: "4")[$4$] feet and #math.equation(block: false, alt: "9")[$9$] feet. How long is the third side? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A triangle with sides labeled 4 ft and 9 ft, and a third side 'c'. The perimeter of the triangle is given as P = 24 ft, implying the missing side 'c' is 11 ft.], alt: "A triangle with sides labeled 4 ft and 9 ft, and a third side 'c'. The perimeter of the triangle is given as P = 24 ft, implying the missing side 'c' is 11 ft.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [length of the third side of a triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[c] = the third side], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute in the given information.], [#linebreak() #figure(figph[Mathematical problem illustrating variable substitution: P = a + b + c, with 24 = 4 + 9 + c provided.], alt: "Mathematical problem illustrating variable substitution: P = a + b + c, with 24 = 4 + 9 + c provided.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A simple algebraic equation is displayed on a white background. The first line reads '24 = 13 + c' and the second line shows its solution as '11 = c', indicating that the value of 'c' is 11.], alt: "A simple algebraic equation is displayed on a white background. The first line reads '24 = 13 + c' and the second line shows its solution as '11 = c', indicating that the value of 'c' is 11.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[An image illustrating a mathematical check: P = a + b + c. The calculation verifies if 24 equals 4 + 9 + 11, concluding with a confirmed equality of 24 = 24, marked by a checkmark.], alt: "An image illustrating a mathematical check: P = a + b + c. The calculation verifies if 24 equals 4 + 9 + 11, concluding with a confirmed equality of 24 = 24, marked by a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The third side is 11 feet long.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The perimeter of a triangular garden is #math.equation(block: false, alt: "48")[$48$] feet. The lengths of two sides are #math.equation(block: false, alt: "18")[$18$] feet and #math.equation(block: false, alt: "22")[$22$] feet. How long is the third side? #solutionbox[ 8 ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The lengths of two sides of a triangular window are #math.equation(block: false, alt: "7")[$7$] feet and #math.equation(block: false, alt: "5")[$5$] feet. The perimeter is #math.equation(block: false, alt: "18")[$18$] feet. How long is the third side? #solutionbox[ 6 ft ] ] #examplebox("Example 11")[][ The area of a triangular church window is #math.equation(block: false, alt: "90")[$90$] square meters. The base of the window is #math.equation(block: false, alt: "15")[$15$] meters. What is the window’s height? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[An isosceles triangle shown with a base of 15 m and an indicated height 'h' perpendicular to the base.], alt: "An isosceles triangle shown with a base of 15 m and an indicated height 'h' perpendicular to the base.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [height of a triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[h] = the height], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute in the given information.], [#linebreak() #figure(figph[The area of a triangle formula A = (1/2) \* b \* h is shown. Below, an example calculates the height, h, with A=90 and b=15, resulting in the equation 90 = (1/2) \* 15 \* h.], alt: "The area of a triangle formula A = (1/2) * b * h is shown. Below, an example calculates the height, h, with A=90 and b=15, resulting in the equation 90 = (1/2) * 15 * h.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[Mathematical equations showing the calculation of 'h', where 90 equals (15/2)h, resulting in the solution h=12.], alt: "Mathematical equations showing the calculation of 'h', where 90 equals (15/2)h, resulting in the solution h=12.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[The image illustrates the verification of a triangle's area using the formula A = (1/2)bh. It confirms that 90 is indeed the correct area when the base is 15 and height is 12, as 90 = (1/2) \* 15 \* 12.], alt: "The image illustrates the verification of a triangle's area using the formula A = (1/2)bh. It confirms that 90 is indeed the correct area when the base is 15 and height is 12, as 90 = (1/2) * 15 * 12.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The height of the triangle is 12 meters.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The area of a triangular painting is #math.equation(block: false, alt: "126")[$126$] square inches. The base is #math.equation(block: false, alt: "18")[$18$] inches. What is the height? #solutionbox[ 14 in. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A triangular tent door has an area of #math.equation(block: false, alt: "15")[$15$] square feet. The height is #math.equation(block: false, alt: "5")[$5$] feet. What is the base? #solutionbox[ 6 ft ] ] ==== Isosceles and Equilateral Triangles Besides the right triangle, some other triangles have special names. A triangle with two sides of equal length is called an #strong[isosceles triangle]. A triangle that has three sides of equal length is called an #strong[equilateral triangle]. shows both types of triangles. #figure(figph[Two triangles are shown. All three sides of the triangle on the left are labeled s. It is labeled “equilateral triangle”. Two sides of the triangle on the right are labeled s. It is labeled “isosceles triangle”.], alt: "Two triangles are shown. All three sides of the triangle on the left are labeled s. It is labeled “equilateral triangle”. Two sides of the triangle on the right are labeled s. It is labeled “isosceles triangle”.", caption: [In an isosceles triangle, two sides have the same length, and the third side is the base. In an equilateral triangle, all three sides have the same length.]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Isosceles and Equilateral Triangles] An #strong[isosceles] triangle has two sides the same length. An #strong[equilateral] triangle has three sides of equal length. ] #examplebox("Example 12")[][ The perimeter of an equilateral triangle is #math.equation(block: false, alt: "93")[$93$] inches. Find the length of each side. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A simple black outline drawing of an equilateral triangle, with each of its three sides labeled with the letter 's', indicating that all sides have equal length.], alt: "A simple black outline drawing of an equilateral triangle, with each of its three sides labeled with the letter 's', indicating that all sides have equal length.", caption: none) #linebreak() Perimeter = 93 in.]), [Step 2. #strong[Identify] what you are looking for.], [length of the sides of an equilateral triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[s] = length of each side], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Mathematical equations illustrate perimeter. P=a+b+c defines the general case, while 93=s+s+s provides a specific example where the perimeter is 93 and the three sides are equal (s).], alt: "Mathematical equations illustrate perimeter. P=a+b+c defines the general case, while 93=s+s+s provides a specific example where the perimeter is 93 and the three sides are equal (s).", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[The image displays a two-step algebraic solution. The top line shows the equation 93 = 3s, and the bottom line shows the solution s = 31, indicating that both sides of the initial equation were divided by 3.], alt: "The image displays a two-step algebraic solution. The top line shows the equation 93 = 3s, and the bottom line shows the solution s = 31, indicating that both sides of the initial equation were divided by 3.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A perfectly balanced equilateral triangle, with each of its three sides clearly marked as 31.], alt: "A perfectly balanced equilateral triangle, with each of its three sides clearly marked as 31.", caption: none) #linebreak() #figure(figph[A math problem demonstrating 93 is equal to 31 + 31 + 31, which simplifies to 93 = 93, confirmed with a checkmark.], alt: "A math problem demonstrating 93 is equal to 31 + 31 + 31, which simplifies to 93 = 93, confirmed with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [Each side is 31 inches.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the length of each side of an equilateral triangle with perimeter #math.equation(block: false, alt: "39")[$39$] inches. #solutionbox[ 13 in. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Find the length of each side of an equilateral triangle with perimeter #math.equation(block: false, alt: "51")[$51$] centimeters. #solutionbox[ 17 cm ] ] #examplebox("Example 13")[][ Arianna has #math.equation(block: false, alt: "156")[$156$] inches of beading to use as trim around a scarf. The scarf will be an isosceles triangle with a base of #linebreak() #math.equation(block: false, alt: "60")[$60$] inches. How long can she make the two equal sides? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A diagram of an isosceles triangle with two equal sides labeled 's' and the base labeled '60 in'.], alt: "A diagram of an isosceles triangle with two equal sides labeled 's' and the base labeled '60 in'.", caption: none) #linebreak() #emph[P] = 156 in.]), [Step 2. #strong[Identify] what you are looking for.], [the lengths of the two equal sides], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[s] = the length of each side], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute in the given information.], [#linebreak() #figure(figph[A two-row mathematical equation showing P = a + b + c in the first row, and 156 = s + 60 + s in the second row, representing a perimeter calculation with a known total and variable sides.], alt: "A two-row mathematical equation showing P = a + b + c in the first row, and 156 = s + 60 + s in the second row, representing a perimeter calculation with a known total and variable sides.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[An image displaying the step-by-step solution to the linear equation 156 = 2s + 60, which proceeds to 96 = 2s, and finally resolves to 48 = s.], alt: "An image displaying the step-by-step solution to the linear equation 156 = 2s + 60, which proceeds to 96 = 2s, and finally resolves to 48 = s.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[Mathematical verification of an equation. The general formula p = a + b + c is applied to check if 156 = 48 + 60 + 48. The calculation confirms 156 = 156 with a checkmark.], alt: "Mathematical verification of an equation. The general formula p = a + b + c is applied to check if 156 = 48 + 60 + 48. The calculation confirms 156 = 156 with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [Arianna can make each of the two equal sides 48 inches long.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A backyard deck is in the shape of an isosceles triangle with a base of #math.equation(block: false, alt: "20")[$20$] feet. The perimeter of the deck is #math.equation(block: false, alt: "48")[$48$] feet. How long is each of the equal sides of the deck? #solutionbox[ 14 ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A boat’s sail is an isosceles triangle with base of #math.equation(block: false, alt: "8")[$8$] meters. The perimeter is #math.equation(block: false, alt: "22")[$22$] meters. How long is each of the equal sides of the sail? #solutionbox[ 7 m ] ] === Use the Properties of Trapezoids A #strong[trapezoid] is four-sided figure, a #emph[quadrilateral], with two sides that are parallel and two sides that are not. The parallel sides are called the bases. We call the length of the smaller base #math.equation(block: false, alt: "b ,")[$b ,$] and the length of the bigger base #math.equation(block: false, alt: "B .")[$B .$] The height, #math.equation(block: false, alt: "h ,")[$h ,$] of a trapezoid is the distance between the two bases as shown. #figure(figph[A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.], alt: "A trapezoid is shown. The top is labeled b and marked as the smaller base. The bottom is labeled B and marked as the larger base. A vertical line forms a right angle with both bases and is marked as h.", caption: [A trapezoid has a larger base, #math.equation(block: false, alt: "B ,")[$B ,$] and a smaller base, #math.equation(block: false, alt: "b .")[$b .$] The height #math.equation(block: false, alt: "h")[$h$] is the distance between the bases.]) The formula for the #strong[area of a trapezoid] is: #math.equation(block: true, alt: "Area sub trapezoid equals the fraction 1 over 2 h open parenthesis b plus B close parenthesis")[$"Area"_("trapezoid") = frac(1, 2) h ( b + B )$]Splitting the trapezoid into two triangles may help us understand the formula. The area of the trapezoid is the sum of the areas of the two triangles. #figure(figph[An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.], alt: "An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner.", caption: [Splitting a trapezoid into two triangles may help you understand the formula for its area.]) The height of the trapezoid is also the height of each of the two triangles. #figure(figph[An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.], alt: "An image of a trapezoid is shown. The top is labeled with a small b, the bottom with a big B. A diagonal is drawn in from the upper left corner to the bottom right corner. There is an arrow pointing to a second trapezoid. The upper right-hand side of the trapezoid forms a blue triangle, with the height of the trapezoid drawn in as a dotted line. The lower left-hand side of the trapezoid forms a red triangle, with the height of the trapezoid drawn in as a dotted line.", caption: none) The formula for the area of a trapezoid is #figure(figph[This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).], alt: "This image shows the formula for the area of a trapezoid and says “area of trapezoid equals one-half h times smaller base b plus larger base B).", caption: none) If we distribute, we get, #figure(figph[The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.], alt: "The top line says area of trapezoid equals one-half times blue little b times h plus one-half times red big B times h. Below this is area of trapezoid equals A sub blue triangle plus A sub red triangle.", caption: none) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Trapezoids] - A trapezoid has four sides. - Two of its sides are parallel and two sides are not. - The area, #math.equation(block: false, alt: "A ,")[$A ,$] of a trapezoid is #math.equation(block: false, alt: "A equals the fraction 1 over 2 h open parenthesis b plus B close parenthesis")[$"A" = frac(1, 2) h ( b + B )$]. ] #examplebox("Example 14")[][ Find the area of a trapezoid whose height is 6 inches and whose bases are #math.equation(block: false, alt: "14")[$14$] and #math.equation(block: false, alt: "11")[$11$] inches. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A trapezoid is displayed with a top base of 14 inches, a bottom base of 11 inches, and a height of 6 inches, indicated by a dashed line and right angle symbols.], alt: "A trapezoid is displayed with a top base of 14 inches, a bottom base of 11 inches, and a height of 6 inches, indicated by a dashed line and right angle symbols.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the area of the trapezoid], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "A equals the area")[$A = "the area"$]], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Illustrates the area formula for a trapezoid, A = 1/2 \* h \* (b + B), and its application with h=6, b=11, B=14.], alt: "Illustrates the area formula for a trapezoid, A = 1/2 * h * (b + B), and its application with h=6, b=11, B=14.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[This image demonstrates the calculation of area A. Starting with A = 1/2 \* 6 \* 25, the equation is simplified to A = 3 \* 25, which ultimately results in an area of A = 75 square inches.], alt: "This image demonstrates the calculation of area A. Starting with A = 1/2 * 6 * 25, the equation is simplified to A = 3 * 25, which ultimately results in an area of A = 75 square inches.", caption: none)], [Step 6. #strong[Check:] Is this answer reasonable?], [], )) If we draw a rectangle around the trapezoid that has the same big base #math.equation(block: false, alt: "B")[$B$] and a height #math.equation(block: false, alt: "h ,")[$h ,$] its area should be greater than that of the trapezoid. If we draw a rectangle inside the trapezoid that has the same little base #math.equation(block: false, alt: "b")[$b$] and a height #math.equation(block: false, alt: "h ,")[$h ,$] its area should be smaller than that of the trapezoid. #figure(figph[A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.], alt: "A table is shown with 3 columns and 4 rows. The first column has an image of a trapezoid with a rectangle drawn around it in red. The larger base of the trapezoid is labeled 14 and is the same as the base of the rectangle. The height of the trapezoid is labeled 6 and is the same as the height of the rectangle. The smaller base of the trapezoid is labeled 11. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 14 times 6. Below is A sub rectangle equals 84 square inches. The second column has an image of a trapezoid. The larger base is labeled 14, the smaller base is labeled 11, and the height is labeled 6. Below this is A sub trapezoid equals one-half times h times parentheses little b plus big B. Below this is A sub trapezoid equals one-half times 6 times parentheses 11 plus 14. Below this is A sub trapezoid equals 75 square inches. The third column has an image of a trapezoid with a red rectangle drawn inside of it. The height is labeled 6. Below this is A sub rectangle equals b times h. Below is A sub rectangle equals 11 times 6. Below is A sub rectangle equals 66 square inches.", caption: none) The area of the larger rectangle is #math.equation(block: false, alt: "84")[$84$] square inches and the area of the smaller rectangle is #math.equation(block: false, alt: "66")[$66$] square inches. So it makes sense that the area of the trapezoid is between #math.equation(block: false, alt: "84")[$84$] and #math.equation(block: false, alt: "66")[$66$] square inches Step 7. #strong[Answer] the question. The area of the trapezoid is #math.equation(block: false, alt: "75")[$75$] square inches. ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The height of a trapezoid is #math.equation(block: false, alt: "14")[$14$] yards and the bases are #math.equation(block: false, alt: "7")[$7$] and #math.equation(block: false, alt: "16")[$16$] yards. What is the area? #solutionbox[ 161 sq. yd ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The height of a trapezoid is #math.equation(block: false, alt: "18")[$18$] centimeters and the bases are #math.equation(block: false, alt: "17")[$17$] and #math.equation(block: false, alt: "8")[$8$] centimeters. What is the area? #solutionbox[ 225 sq. cm ] ] #examplebox("Example 15")[][ Find the area of a trapezoid whose height is #math.equation(block: false, alt: "5")[$5$] feet and whose bases are #math.equation(block: false, alt: "10.3")[$10.3$] and #math.equation(block: false, alt: "13.7")[$13.7$] feet. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A white background shows a trapezoid with its top base measuring 10.3 ft, its bottom base measuring 13.7 ft, and its height measuring 5 ft. A dashed vertical line indicates the height, with right angle symbols.], alt: "A white background shows a trapezoid with its top base measuring 10.3 ft, its bottom base measuring 13.7 ft, and its height measuring 5 ft. A dashed vertical line indicates the height, with right angle symbols.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [the area of the trapezoid], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[A] = the area], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Formula for the area of a trapezoid (A = 1/2 \* h \* (b + B)) and its application with specific values: h=5, b=10.3, B=13.7.], alt: "Formula for the area of a trapezoid (A = 1/2 * h * (b + B)) and its application with specific values: h=5, b=10.3, B=13.7.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical calculation is shown where the area (A) is determined. Starting with A = (1/2) \* 5 \* 24, the steps lead to A = 12 \* 5, and finally A = 60 square feet, presented in three lines on a white background.], alt: "A mathematical calculation is shown where the area (A) is determined. Starting with A = (1/2) * 5 * 24, the steps lead to A = 12 * 5, and finally A = 60 square feet, presented in three lines on a white background.", caption: none)], [Step 6. #strong[Check:] Is this answer reasonable? #linebreak() The area of the trapezoid should be less than the area of a rectangle with base 13.7 and height 5, but more than the area of a rectangle with base 10.3 and height 5. #linebreak() #figure(figph[An image of a trapezoid is shown with a red rectangle drawn around it. The larger base of the trapezoid is labeled 13.7 ft. and is the same as the base of the rectangle. The height of both the trapezoid and the rectangle is 5 ft. Next to this is an image of a trapezoid with a black rectangle drawn inside it. The smaller base of the trapezoid is labeled 10.3 ft. and is the same as the base of the rectangle. Below the images is A sub red rectangle is greater than A sub trapezoid is greater than A sub rectangle. Below this is 68.5, 60, and 51.5.], alt: "An image of a trapezoid is shown with a red rectangle drawn around it. The larger base of the trapezoid is labeled 13.7 ft. and is the same as the base of the rectangle. The height of both the trapezoid and the rectangle is 5 ft. Next to this is an image of a trapezoid with a black rectangle drawn inside it. The smaller base of the trapezoid is labeled 10.3 ft. and is the same as the base of the rectangle. Below the images is A sub red rectangle is greater than A sub trapezoid is greater than A sub rectangle. Below this is 68.5, 60, and 51.5.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The area of the trapezoid is 60 square feet.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The height of a trapezoid is #math.equation(block: false, alt: "7")[$7$] centimeters and the bases are #math.equation(block: false, alt: "4.6")[$4.6$] and #math.equation(block: false, alt: "7.4")[$7.4$] centimeters. What is the area? #solutionbox[ 42 sq. cm ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The height of a trapezoid is #math.equation(block: false, alt: "9")[$9$] meters and the bases are #math.equation(block: false, alt: "6.2")[$6.2$] and #math.equation(block: false, alt: "7.8")[$7.8$] meters. What is the area? #solutionbox[ 63 sq. m ] ] #examplebox("Example 16")[][ Vinny has a garden that is shaped like a trapezoid. The trapezoid has a height of #math.equation(block: false, alt: "3.4")[$3.4$] yards and the bases are #math.equation(block: false, alt: "8.2")[$8.2$] and #math.equation(block: false, alt: "5.6")[$5.6$] yards. How many square yards will be available to plant? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A right trapezoid is shown with a top base of 5.6 yd, a bottom base of 8.2 yd, and a height of 3.4 yd, indicated by right angle markings on the vertical side.], alt: "A right trapezoid is shown with a top base of 5.6 yd, a bottom base of 8.2 yd, and a height of 3.4 yd, indicated by right angle markings on the vertical side.", caption: none)], [Step 2. #strong[Identify] what you are looking for.], [the area of a trapezoid], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[A] = the area], [Step 4.#strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Formula for the area of a trapezoid, A = (1/2) \* h \* (b + B), with h=3.4, b=5.6, and B=8.2 substituted in the second line.], alt: "Formula for the area of a trapezoid, A = (1/2) * h * (b + B), with h=3.4, b=5.6, and B=8.2 substituted in the second line.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical calculation showing A equals one-half times 3.4 times 13.8, resulting in A equals 23.46 square yards.], alt: "A mathematical calculation showing A equals one-half times 3.4 times 13.8, resulting in A equals 23.46 square yards.", caption: none)], [Step 6. #strong[Check:] Is this answer reasonable? #linebreak() Yes. The area of the trapezoid is less than the area of a rectangle with a base of 8.2 yd and height 3.4 yd, but more than the area of a rectangle with base 5.6 yd and height 3.4 yd. #linebreak() #linebreak() #figure(figph[This image is a table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has “equals 27.88 yards squared”. The center column includes the formula of a trapezoid and says Area of a trapezoid equals one half times 3.5 yards in parentheses times 5.8 plus 8.2 in parentheses. Under this it has “equals 23.46 yards squared”. In the third column it it has the formula the area of a rectangle equals base times height. Under this it has equals 5.6 in parentheses times 3.4 in parentheses. Under this it has “equals 19.04 yards squared.” In the second row, centered from left to right it has “Area of a rectangle” and a “greater than” sign, “Area of a trapezoid” and a greater than sign and “area of a rectangle”. Under Area of a rectangle it has 27.88, then 23.46 under “area of a trapezoid”, then 19.04 under “area of a rectangle”.], alt: "This image is a table with two rows. the first row is split into three columns. The first column is the formula Area of a rectangle equals base times height. On the next line under this it has numbers plugged into the formula; the base, 8.2 in parentheses times the height 3.4 in parentheses. Under this is it has “equals 27.88 yards squared”. The center column includes the formula of a trapezoid and says Area of a trapezoid equals one half times 3.5 yards in parentheses times 5.8 plus 8.2 in parentheses. Under this it has “equals 23.46 yards squared”. In the third column it it has the formula the area of a rectangle equals base times height. Under this it has equals 5.6 in parentheses times 3.4 in parentheses. Under this it has “equals 19.04 yards squared.” In the second row, centered from left to right it has “Area of a rectangle” and a “greater than” sign, “Area of a trapezoid” and a greater than sign and “area of a rectangle”. Under Area of a rectangle it has 27.88, then 23.46 under “area of a trapezoid”, then 19.04 under “area of a rectangle”.", caption: none)], [], [Step 7. #strong[Answer] the question.], [Vinny has 23.46 square yards in which he can plant.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Lin wants to sod his lawn, which is shaped like a trapezoid. The bases are #math.equation(block: false, alt: "10.8")[$10.8$] yards and #math.equation(block: false, alt: "6.7")[$6.7$] yards, and the height is #math.equation(block: false, alt: "4.6")[$4.6$] yards. How many square yards of sod does he need? #solutionbox[ 40.25 sq. yd ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Kira wants cover his patio with concrete pavers. If the patio is shaped like a trapezoid whose bases are #math.equation(block: false, alt: "18")[$18$] feet and #math.equation(block: false, alt: "14")[$14$] feet and whose height is #math.equation(block: false, alt: "15")[$15$] feet, how many square feet of pavers will he need? #solutionbox[ 240 sq. ft ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24perirect")[Perimeter of a Rectangle] - #link("https://www.openstax.org/l/24arearect")[Area of a Rectangle] - #link("https://www.openstax.org/l/24periareaform")[Perimeter and Area Formulas] - #link("https://www.openstax.org/l/24areatri")[Area of a Triangle] - #link("https://www.openstax.org/l/24areatrifract")[Area of a Triangle with Fractions] - #link("https://www.openstax.org/l/24areatrap")[Area of a Trapezoid] ] === Key Concepts - #emph[Properties of Rectangles] - Rectangles have four sides and four right (90°) angles. - The lengths of opposite sides are equal. - The perimeter, #math.equation(block: false, alt: "P")[$P$], of a rectangle is the sum of twice the length and twice the width. - #math.equation(block: false, alt: "P equals 2 L plus 2 W")[$P = 2 L + 2 W$] - The area, #math.equation(block: false, alt: "A")[$A$], of a rectangle is the length times the width. - #math.equation(block: false, alt: "A equals L times W")[$A = L ⋅ W$] - #emph[Triangle Properties] - For any triangle #math.equation(block: false, alt: "Δ A B C")[$Δ A B C$], the sum of the measures of the angles is 180°. - #math.equation(block: false, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180°")[$m ∠ A + m ∠ B + m ∠ C = 180°$] - The perimeter of a triangle is the sum of the lengths of the sides. - #math.equation(block: false, alt: "P equals a plus b plus c")[$P = a + b + c$] - The area of a triangle is one-half the base, b, times the height, h. - #math.equation(block: false, alt: "A equals the fraction 1 over 2 b h")[$A = frac(1, 2) b h$] ==== Practice Makes Perfect #strong[Understand Linear, Square, and Cubic Measure] In the following exercises, determine whether you would measure each item using linear, square, or cubic units. amount of water in a fish tank #solutionbox[ cubic ] length of dental floss living area of an apartment #solutionbox[ square ] floor space of a bathroom tile height of a doorway #solutionbox[ linear ] capacity of a truck trailer In the following exercises, find the ⓐ perimeter and ⓑ area of each figure. Assume each side of the square is #math.equation(block: false, alt: "1")[$1$] cm. #figure(figph[A rectangle is shown comprised of 4 squares forming a horizontal line.], alt: "A rectangle is shown comprised of 4 squares forming a horizontal line.", caption: none) #solutionbox[ + ⓐ 10 cm + ⓑ 4 sq. cm ] #figure(figph[A rectangle is shown comprised of 3 squares forming a vertical line.], alt: "A rectangle is shown comprised of 3 squares forming a vertical line.", caption: none) #figure(figph[Three squares are shown. There is one on the bottom left, one on the bottom right, and one on the top right.], alt: "Three squares are shown. There is one on the bottom left, one on the bottom right, and one on the top right.", caption: none) #solutionbox[ + ⓐ 8 cm + ⓑ 3 sq. cm ] #figure(figph[Four squares are shown. Three form a horizontal line, and there is one above the center square.], alt: "Four squares are shown. Three form a horizontal line, and there is one above the center square.", caption: none) #figure(figph[Five squares are shown. There are three forming a horizontal line across the top and two underneath the two on the right.], alt: "Five squares are shown. There are three forming a horizontal line across the top and two underneath the two on the right.", caption: none) #solutionbox[ + ⓐ 10 cm + ⓑ 5 sq. cm ] #figure(figph[A square is shown. It is comprised of nine smaller squares.], alt: "A square is shown. It is comprised of nine smaller squares.", caption: none) #strong[Use the Properties of Rectangles] In the following exercises, find the ⓐ perimeter and ⓑ area of each rectangle. The length of a rectangle is #math.equation(block: false, alt: "85")[$85$] feet and the width is #math.equation(block: false, alt: "45")[$45$] feet. #solutionbox[ + ⓐ 260 ft + ⓑ 3825 sq. ft ] The length of a rectangle is #math.equation(block: false, alt: "26")[$26$] inches and the width is #math.equation(block: false, alt: "58")[$58$] inches. A rectangular room is #math.equation(block: false, alt: "15")[$15$] feet wide by #math.equation(block: false, alt: "14")[$14$] feet long. #solutionbox[ + ⓐ 58 ft + ⓑ 210 sq. ft ] A driveway is in the shape of a rectangle #math.equation(block: false, alt: "20")[$20$] feet wide by #math.equation(block: false, alt: "35")[$35$] feet long. In the following exercises, solve. Find the length of a rectangle with perimeter #math.equation(block: false, alt: "124")[$124$] inches and width #math.equation(block: false, alt: "38")[$38$] inches. #solutionbox[ 24 inches ] Find the length of a rectangle with perimeter #math.equation(block: false, alt: "20.2")[$20.2$] yards and width of #math.equation(block: false, alt: "7.8")[$7.8$] yards. Find the width of a rectangle with perimeter #math.equation(block: false, alt: "92")[$92$] meters and length #math.equation(block: false, alt: "19")[$19$] meters. #solutionbox[ 27 meters ] Find the width of a rectangle with perimeter #math.equation(block: false, alt: "16.2")[$16.2$] meters and length #math.equation(block: false, alt: "3.2")[$3.2$] meters. The area of a rectangle is #math.equation(block: false, alt: "414")[$414$] square meters. The length is #math.equation(block: false, alt: "18")[$18$] meters. What is the width? #solutionbox[ 23 m ] The area of a rectangle is #math.equation(block: false, alt: "782")[$782$] square centimeters. The width is #math.equation(block: false, alt: "17")[$17$] centimeters. What is the length? The length of a rectangle is #math.equation(block: false, alt: "9")[$9$] inches more than the width. The perimeter is #math.equation(block: false, alt: "46")[$46$] inches. Find the length and the width. #solutionbox[ 7 in., 16 in. ] The width of a rectangle is #math.equation(block: false, alt: "8")[$8$] inches more than the length. The perimeter is #math.equation(block: false, alt: "52")[$52$] inches. Find the length and the width. The perimeter of a rectangle is #math.equation(block: false, alt: "58")[$58$] meters. The width of the rectangle is #math.equation(block: false, alt: "5")[$5$] meters less than the length. Find the length and the width of the rectangle. #solutionbox[ 17 m, 12 m ] The perimeter of a rectangle is #math.equation(block: false, alt: "62")[$62$] feet. The width is #math.equation(block: false, alt: "7")[$7$] feet less than the length. Find the length and the width. The width of the rectangle is #math.equation(block: false, alt: "0.7")[$0.7$] meters less than the length. The perimeter of a rectangle is #math.equation(block: false, alt: "52.6")[$52.6$] meters. Find the dimensions of the rectangle. #solutionbox[ 13.5 m, 12.8 m ] The length of the rectangle is #math.equation(block: false, alt: "1.1")[$1.1$] meters less than the width. The perimeter of a rectangle is #math.equation(block: false, alt: "49.4")[$49.4$] meters. Find the dimensions of the rectangle. The perimeter of a rectangle of #math.equation(block: false, alt: "150")[$150$] feet. The length of the rectangle is twice the width. Find the length and width of the rectangle. #solutionbox[ 25 ft, 50 ft ] The length of a rectangle is three times the width. The perimeter is #math.equation(block: false, alt: "72")[$72$] feet. Find the length and width of the rectangle. The length of a rectangle is #math.equation(block: false, alt: "3")[$3$] meters less than twice the width. The perimeter is #math.equation(block: false, alt: "36")[$36$] meters. Find the length and width. #solutionbox[ #emph[l] = 11 m, #emph[w] = 7 m ] The length of a rectangle is #math.equation(block: false, alt: "5")[$5$] inches more than twice the width. The perimeter is #math.equation(block: false, alt: "34")[$34$] inches. Find the length and width. The width of a rectangular window is #math.equation(block: false, alt: "24")[$24$] inches. The area is #math.equation(block: false, alt: "624")[$624$] square inches. What is the length? #solutionbox[ 26 in. ] The length of a rectangular poster is #math.equation(block: false, alt: "28")[$28$] inches. The area is #math.equation(block: false, alt: "1316")[$1316$] square inches. What is the width? The area of a rectangular roof is #math.equation(block: false, alt: "2310")[$2310$] square meters. The length is #math.equation(block: false, alt: "42")[$42$] meters. What is the width? #solutionbox[ 55 m ] The area of a rectangular tarp is #math.equation(block: false, alt: "132")[$132$] square feet. The width is #math.equation(block: false, alt: "12")[$12$] feet. What is the length? The perimeter of a rectangular courtyard is #math.equation(block: false, alt: "160")[$160$] feet. The length is #math.equation(block: false, alt: "10")[$10$] feet more than the width. Find the length and the width. #solutionbox[ 35 ft, 45 ft ] The perimeter of a rectangular painting is #math.equation(block: false, alt: "306")[$306$] centimeters. The length is #math.equation(block: false, alt: "17")[$17$] centimeters more than the width. Find the length and the width. The width of a rectangular window is #math.equation(block: false, alt: "40")[$40$] inches less than the height. The perimeter of the doorway is #math.equation(block: false, alt: "224")[$224$] inches. Find the length and the width. #solutionbox[ 76 in., 36 in. ] The width of a rectangular playground is #math.equation(block: false, alt: "7")[$7$] meters less than the length. The perimeter of the playground is #math.equation(block: false, alt: "46")[$46$] meters. Find the length and the width. #strong[Use the Properties of Triangles] In the following exercises, solve using the properties of triangles. Find the area of a triangle with base #math.equation(block: false, alt: "12")[$12$] inches and height #math.equation(block: false, alt: "5")[$5$] inches. #solutionbox[ 30 sq. in. ] Find the area of a triangle with base #math.equation(block: false, alt: "45")[$45$] centimeters and height #math.equation(block: false, alt: "30")[$30$] centimeters. Find the area of a triangle with base #math.equation(block: false, alt: "8.3")[$8.3$] meters and height #math.equation(block: false, alt: "6.1")[$6.1$] meters. #solutionbox[ 25.315 sq. m ] Find the area of a triangle with base #math.equation(block: false, alt: "24.2")[$24.2$] feet and height #math.equation(block: false, alt: "20.5")[$20.5$] feet. A triangular flag has base of #math.equation(block: false, alt: "1")[$1$] foot and height of #math.equation(block: false, alt: "1.5")[$1.5$] feet. What is its area? #solutionbox[ 0.75 sq. ft ] A triangular window has base of #math.equation(block: false, alt: "8")[$8$] feet and height of #math.equation(block: false, alt: "6")[$6$] feet. What is its area? If a triangle has sides of #math.equation(block: false, alt: "6")[$6$] feet and #math.equation(block: false, alt: "9")[$9$] feet and the perimeter is #math.equation(block: false, alt: "23")[$23$] feet, how long is the third side? #solutionbox[ 8 ft ] If a triangle has sides of #math.equation(block: false, alt: "14")[$14$] centimeters and #math.equation(block: false, alt: "18")[$18$] centimeters and the perimeter is #math.equation(block: false, alt: "49")[$49$] centimeters, how long is the third side? What is the base of a triangle with an area of #math.equation(block: false, alt: "207")[$207$] square inches and height of #math.equation(block: false, alt: "18")[$18$] inches? #solutionbox[ 23 in. ] What is the height of a triangle with an area of #math.equation(block: false, alt: "893")[$893$] square inches and base of #math.equation(block: false, alt: "38")[$38$] inches? The perimeter of a triangular reflecting pool is #math.equation(block: false, alt: "36")[$36$] yards. The lengths of two sides are #math.equation(block: false, alt: "10")[$10$] yards and #math.equation(block: false, alt: "15")[$15$] yards. How long is the third side? #solutionbox[ 11 yd ] A triangular courtyard has perimeter of #math.equation(block: false, alt: "120")[$120$] meters. The lengths of two sides are #math.equation(block: false, alt: "30")[$30$] meters and #math.equation(block: false, alt: "50")[$50$] meters. How long is the third side? An isosceles triangle has a base of #math.equation(block: false, alt: "20")[$20$] centimeters. If the perimeter is #math.equation(block: false, alt: "76")[$76$] centimeters, find the length of each of the other sides. #solutionbox[ 28 cm ] An isosceles triangle has a base of #math.equation(block: false, alt: "25")[$25$] inches. If the perimeter is #math.equation(block: false, alt: "95")[$95$] inches, find the length of each of the other sides. Find the length of each side of an equilateral triangle with a perimeter of #math.equation(block: false, alt: "51")[$51$] yards. #solutionbox[ 17 yd ] Find the length of each side of an equilateral triangle with a perimeter of #math.equation(block: false, alt: "54")[$54$] meters. The perimeter of an equilateral triangle is #math.equation(block: false, alt: "18")[$18$] meters. Find the length of each side. #solutionbox[ 6 m ] The perimeter of an equilateral triangle is #math.equation(block: false, alt: "42")[$42$] miles. Find the length of each side. The perimeter of an isosceles triangle is #math.equation(block: false, alt: "42")[$42$] feet. The length of the shortest side is #math.equation(block: false, alt: "12")[$12$] feet. Find the length of the other two sides. #solutionbox[ 15 ft ] The perimeter of an isosceles triangle is #math.equation(block: false, alt: "83")[$83$] inches. The length of the shortest side is #math.equation(block: false, alt: "24")[$24$] inches. Find the length of the other two sides. A dish is in the shape of an equilateral triangle. Each side is #math.equation(block: false, alt: "8")[$8$] inches long. Find the perimeter. #solutionbox[ 24 in. ] A floor tile is in the shape of an equilateral triangle. Each side is #math.equation(block: false, alt: "1.5")[$1.5$] feet long. Find the perimeter. A road sign in the shape of an isosceles triangle has a base of #math.equation(block: false, alt: "36")[$36$] inches. If the perimeter is #math.equation(block: false, alt: "91")[$91$] inches, find the length of each of the other sides. #solutionbox[ 27.5 in. ] A scarf in the shape of an isosceles triangle has a base of #math.equation(block: false, alt: "0.75")[$0.75$] meters. If the perimeter is #math.equation(block: false, alt: "2")[$2$] meters, find the length of each of the other sides. The perimeter of a triangle is #math.equation(block: false, alt: "39")[$39$] feet. One side of the triangle is #math.equation(block: false, alt: "1")[$1$] foot longer than the second side. The third side is #math.equation(block: false, alt: "2")[$2$] feet longer than the second side. Find the length of each side. #solutionbox[ 12 ft, 13 ft, 14 ft ] The perimeter of a triangle is #math.equation(block: false, alt: "35")[$35$] feet. One side of the triangle is #math.equation(block: false, alt: "5")[$5$] feet longer than the second side. The third side is #math.equation(block: false, alt: "3")[$3$] feet longer than the second side. Find the length of each side. One side of a triangle is twice the smallest side. The third side is #math.equation(block: false, alt: "5")[$5$] feet more than the shortest side. The perimeter is #math.equation(block: false, alt: "17")[$17$] feet. Find the lengths of all three sides. #solutionbox[ 3 ft, 6 ft, 8 ft ] One side of a triangle is three times the smallest side. The third side is #math.equation(block: false, alt: "3")[$3$] feet more than the shortest side. The perimeter is #math.equation(block: false, alt: "13")[$13$] feet. Find the lengths of all three sides. #strong[Use the Properties of Trapezoids] In the following exercises, solve using the properties of trapezoids. The height of a trapezoid is #math.equation(block: false, alt: "12")[$12$] feet and the bases are #math.equation(block: false, alt: "9")[$9$] and #math.equation(block: false, alt: "15")[$15$] feet. What is the area? #solutionbox[ 144 sq. ft ] The height of a trapezoid is #math.equation(block: false, alt: "24")[$24$] yards and the bases are #math.equation(block: false, alt: "18")[$18$] and #math.equation(block: false, alt: "30")[$30$] yards. What is the area? Find the area of a trapezoid with a height of #math.equation(block: false, alt: "51")[$51$] meters and bases of #math.equation(block: false, alt: "43")[$43$] and #math.equation(block: false, alt: "67")[$67$] meters. #solutionbox[ 2805 sq. m ] Find the area of a trapezoid with a height of #math.equation(block: false, alt: "62")[$62$] inches and bases of #math.equation(block: false, alt: "58")[$58$] and #math.equation(block: false, alt: "75")[$75$] inches. The height of a trapezoid is #math.equation(block: false, alt: "15")[$15$] centimeters and the bases are #math.equation(block: false, alt: "12.5")[$12.5$] and #math.equation(block: false, alt: "18.3")[$18.3$] centimeters. What is the area? #solutionbox[ 231 sq. cm ] The height of a trapezoid is #math.equation(block: false, alt: "48")[$48$] feet and the bases are #math.equation(block: false, alt: "38.6")[$38.6$] and #math.equation(block: false, alt: "60.2")[$60.2$] feet. What is the area? Find the area of a trapezoid with a height of #math.equation(block: false, alt: "4.2")[$4.2$] meters and bases of #math.equation(block: false, alt: "8.1")[$8.1$] and #math.equation(block: false, alt: "5.5")[$5.5$] meters. #solutionbox[ 28.56 sq. m ] Find the area of a trapezoid with a height of #math.equation(block: false, alt: "32.5")[$32.5$] centimeters and bases of #math.equation(block: false, alt: "54.6")[$54.6$] and #math.equation(block: false, alt: "41.4")[$41.4$] centimeters. Laurel is making a banner shaped like a trapezoid. The height of the banner is #math.equation(block: false, alt: "3")[$3$] feet and the bases are #math.equation(block: false, alt: "4")[$4$] and #math.equation(block: false, alt: "5")[$5$] feet. What is the area of the banner? #solutionbox[ 13.5 sq. ft ] Niko wants to tile the floor of his bathroom. The floor is shaped like a trapezoid with width #math.equation(block: false, alt: "5")[$5$] feet and lengths #math.equation(block: false, alt: "5")[$5$] feet and #math.equation(block: false, alt: "8")[$8$] feet. What is the area of the floor? Theresa needs a new top for her kitchen counter. The counter is shaped like a trapezoid with width #math.equation(block: false, alt: "18.5")[$18.5$] inches and lengths #math.equation(block: false, alt: "62")[$62$] and #math.equation(block: false, alt: "50")[$50$] inches. What is the area of the counter? #solutionbox[ 1036 sq. in. ] Elena is knitting a scarf. The scarf will be shaped like a trapezoid with width #math.equation(block: false, alt: "8")[$8$] inches and lengths #math.equation(block: false, alt: "48.2")[$48.2$] inches and #math.equation(block: false, alt: "56.2")[$56.2$] inches. What is the area of the scarf? ==== Everyday Math #strong[Fence] Jose just removed the children’s playset from his back yard to make room for a rectangular garden. He wants to put a fence around the garden to keep out the dog. He has a #math.equation(block: false, alt: "50")[$50$] foot roll of fence in his garage that he plans to use. To fit in the backyard, the width of the garden must be #math.equation(block: false, alt: "10")[$10$] feet. How long can he make the other side if he wants to use the entire roll of fence? #solutionbox[ 15 ft ] #strong[Gardening] Lupita wants to fence in her tomato garden. The garden is rectangular and the length is twice the width. It will take #math.equation(block: false, alt: "48")[$48$] feet of fencing to enclose the garden. Find the length and width of her garden. #strong[Fence] Christa wants to put a fence around her triangular flowerbed. The sides of the flowerbed are #math.equation(block: false, alt: "6")[$6$] feet, #math.equation(block: false, alt: "8")[$8$] feet, and #math.equation(block: false, alt: "10")[$10$] feet. The fence costs #math.equation(block: false, alt: "$10")[$\$10$] per foot. How much will it cost for Christa to fence in her flowerbed? #solutionbox[ \$240 ] #strong[Painting] Caleb wants to paint one wall of his attic. The wall is shaped like a trapezoid with height #math.equation(block: false, alt: "8")[$8$] feet and bases #math.equation(block: false, alt: "20")[$20$] feet and #math.equation(block: false, alt: "12")[$12$] feet. The cost of the painting one square foot of wall is about #math.equation(block: false, alt: "$0.05 .")[$\$0.05 .$] About how much will it cost for Caleb to paint the attic wall? #figure(figph[A right trapezoid is shown.], alt: "A right trapezoid is shown.", caption: none) ==== Writing Exercises If you need to put tile on your kitchen floor, do you need to know the perimeter or the area of the kitchen? Explain your reasoning. #solutionbox[ Answers will vary. ] If you need to put a fence around your backyard, do you need to know the perimeter or the area of the backyard? Explain your reasoning. Look at the two figures. #figure(figph[A rectangle is shown on the left. It is labeled as 2 by 8. A square is shown on the right. It is labeled as 4 by 4.], alt: "A rectangle is shown on the left. It is labeled as 2 by 8. A square is shown on the right. It is labeled as 4 by 4.", caption: none) ⓐ Which figure looks like it has the larger area? Which looks like it has the larger perimeter? ⓑ Now calculate the area and perimeter of each figure. Which has the larger area? Which has the larger perimeter? #solutionbox[ Answers will vary. ] The length of a rectangle is #math.equation(block: false, alt: "5")[$5$] feet more than the width. The area is #math.equation(block: false, alt: "50")[$50$] square feet. Find the length and the width. ⓐ Write the equation you would use to solve the problem. ⓑ Why can’t you solve this equation with the methods you learned in the previous chapter? ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment table with geometry skills. The rows list skills like understanding measures and using properties of rectangles, triangles, and trapezoids. Columns are for 'Confidently', 'With some help', and 'No-I don't get it!'], alt: "A self-assessment table with geometry skills. The rows list skills like understanding measures and using properties of rectangles, triangles, and trapezoids. Columns are for 'Confidently', 'With some help', and 'No-I don't get it!'", caption: none) ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?