#set document(title: "1.4 Chapter 1 Summary and Review", 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")) == 1.4#h(0.6em)Chapter 1 Summary and Review === Key Concepts + The sum of the angles in a triangle is #math.equation(block: false, alt: "180 degrees")[$180^(∘)$]. + A right triangle has one angle of #math.equation(block: false, alt: "90 degrees")[$90^(∘)$]. + All of the angles of an equilateral triangle are equal. + The base angles of an isosceles triangle are equal. + Vertical angles are equal. + If parallel lines are intersected by a transversal, the alternate interior angles are equal. Corresponding angles are also equal. + Two triangles are #strong[congruent] if they have exactly the same size and shape. + The #strong[altitude] of an equilateral triangle divides it into two congruent right triangles. + In a #math.equation(block: false, alt: "30 degrees minus 60 degrees minus 90 degrees")[$30^(∘) − 60^(∘) − 90^(∘)$] right triangle, the leg opposite the #math.equation(block: false, alt: "30 degrees")[$30^(∘)$] angle is half the length of the hypotenuse. + Two triangles are #strong[similar] if they have the same shape but not necessarily the same size. The corresponding angles are equal, and the corresponding sides are #strong[proportional]. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Similar Triangles] Two triangles are similar if either + their corresponding angles are equal, or + their corresponding sides are proportional. ] + If two #emph[right] triangles have #emph[one] pair of corresponding acute angles with the same measure, then the triangles are similar. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Distance Formula] The distance #math.equation(block: false, alt: "d")[$d$] between two points #math.equation(block: false, alt: "P sub 1 open parenthesis x sub 1 , y sub 1 close parenthesis")[$P_(1) ( x_(1) , y_(1) )$] and #math.equation(block: false, alt: "P sub 2 open parenthesis x sub 2 , y sub 2 close parenthesis")[$P_(2) ( x_(2) , y_(2) )$] is #math.equation(block: true, alt: "d equals the square root of open parenthesis x sub 2 minus x sub 1 close parenthesis squared plus open parenthesis y sub 2 minus y sub 1 close parenthesis squared")[$d = sqrt(( x_(2) − x_(1) )^(2) + ( y_(2) − y_(1) )^(2))$] ] + Any number that can be written as a quotient of two integers #math.equation(block: false, alt: "the fraction a over b , b not equal to 0 ,")[$display(frac(a, b)) , " " " " b ≠ 0 , " " " "$], is called a #strong[rational number]. The decimal form of a rational number is either a #strong[terminating decimal] or a #strong[repeating decimal]. + An #strong[irrational number] is one that cannot be written as a quotient of two integers #math.equation(block: false, alt: "the fraction a over b , b not equal to 0 ,")[$display(frac(a, b)) , " " " " b ≠ 0 , " " " "$]. We cannot write down an exact decimal equivalent for an irrational number. + A #strong[circle] is the set of all points in a plane that lie at a given distance, called the #strong[radius], from a fixed point called the #strong[center]. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Circle] The equation for a #strong[circle] of radius #math.equation(block: false, alt: "r")[$r$] centered at the origin is #math.equation(block: true, alt: "x squared plus y squared equals r squared")[$x^(2) + y^(2) = r^(2)$] ] + The circle #math.equation(block: false, alt: "x squared plus y squared equals 1")[$x^(2) + y^(2) = 1$], which is centered at the origin and has radius 1 unit, is called the #strong[unit circle]. + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Circumference of a Circle] The #strong[circumference] of a circle of radius #math.equation(block: false, alt: "r")[$r$] is given by #math.equation(block: true, alt: "C equals 2 π r")[$C = 2 π r$] ] + #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Area of a Circle] The #strong[area] of a circle of radius #math.equation(block: false, alt: "r")[$r$] is given by #math.equation(block: true, alt: "A equals π r squared")[$A = π r^(2)$] ] === Chapter 1 Review Problems For Problems 1–4, sketch the triangle described. An isosceles triangle with vertex angle #math.equation(block: false, alt: "100 degrees")[$100^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) An isosceles triangle with base angles of #math.equation(block: false, alt: "75 degrees")[$75^(∘)$] A scalene right triangle #figure(figph[triangle], alt: "triangle", caption: none) A scalene triangle with one obtuse angle For Problems 5-–16, find the unknown angles. #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "α equals β equals γ equals 60 degrees")[$α = β = γ = 60^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "ϕ equals ω equals 79 degrees")[$ϕ = ω = 79^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "θ equals 65 degrees")[$θ = 65^(∘)$], #math.equation(block: true, alt: "ϕ equals 25 degrees")[$ϕ = 25^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "δ equals 30 degrees")[$δ = 30^(∘)$], #math.equation(block: true, alt: "γ equals 60 degrees")[$γ = 60^(∘)$] #figure(figph[square], alt: "square", caption: none) #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "σ equals 39 degrees")[$σ = 39^(∘)$], #math.equation(block: true, alt: "ω equals 79 degrees")[$ω = 79^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) #figure(figph[hexagon], alt: "hexagon", caption: none) #math.equation(block: true, alt: "α equals 51 the fraction 3 over 7 degrees")[$α = 51 attach(frac(3, 7), t: ∘)$], #math.equation(block: true, alt: "β equals 64 the fraction 2 over 7 degrees")[$β = 64 attach(frac(2, 7), t: ∘)$] #figure(figph[octagon], alt: "octagon", caption: none) In Problems 17 and 18, name two congruent triangles and find the unknown quantities. #figure(figph[parallel lines], alt: "parallel lines", caption: none) #math.equation(block: true, alt: "△ A B C ≅ △ E D C")[$upright(△) A B C ≅ upright(△) E D C$], #math.equation(block: true, alt: "α equals 40 degrees")[$α = 40^(∘)$], #math.equation(block: true, alt: "β equals 130 degrees")[$β = 130^(∘)$], #math.equation(block: true, alt: "x equals 32")[$x = 32$] #math.equation(block: false, alt: "P Q R S")[$P Q R S$] is a square #figure(figph[square], alt: "square", caption: none) In Problems 19–22, are the pairs of triangles are similar? Explain why or why not. #figure(figph[triangles], alt: "triangles", caption: none) Yes, three pairs of equal angles #figure(figph[triangles], alt: "triangles", caption: none) #figure(figph[triangles], alt: "triangles", caption: none) Yes, three pairs of equal angles #figure(figph[triangles], alt: "triangles", caption: none) In Problems 23–26, find the unknown side. #figure(figph[parallel lines], alt: "parallel lines", caption: none) 13 #figure(figph[squares], alt: "squares", caption: none) #figure(figph[parallel lines], alt: "parallel lines", caption: none) 18 #figure(figph[squares], alt: "squares", caption: none) In Problems 27–34, solve for #math.equation(block: false, alt: "y")[$y$] in terms of #math.equation(block: false, alt: "x")[$x$]. #figure(figph[triangles], alt: "triangles", caption: none) #math.equation(block: true, alt: "y equals the fraction 5 x over 2")[$y = display(frac(5 x, 2))$] #figure(figph[triangles], alt: "triangles", caption: none) #figure(figph[triangles], alt: "triangles", caption: none) #math.equation(block: true, alt: "y equals the fraction 7 x over 3")[$y = display(frac(7 x, 3))$] #figure(figph[triangles], alt: "triangles", caption: none) #figure(figph[triangles], alt: "triangles", caption: none) #math.equation(block: true, alt: "y equals the fraction x over 3")[$y = display(frac(x, 3))$] #figure(figph[triangles], alt: "triangles", caption: none) #figure(figph[rectangles], alt: "rectangles", caption: none) #math.equation(block: true, alt: "y equals the fraction 12 x over 5")[$y = display(frac(12 x, 5))$] #figure(figph[rectangles], alt: "rectangles", caption: none) In Problems 35 and 36, find angle #math.equation(block: false, alt: "α")[$α$]. The gray lines are horizontal. #figure(figph[triangle], alt: "triangle", caption: none) #math.equation(block: true, alt: "α equals 70 degrees")[$α = 70^(∘)$] #figure(figph[triangle], alt: "triangle", caption: none) For Problems 37–40, make a sketch showing similar triangles, write a proportion, and solve. A 6-foot man stands 12 feet from a lamppost. His shadow is 9 feet long. How tall is the lamppost? #figure(figph[lamppost], alt: "lamppost", caption: none) 14 ft Judy is observing the Mr. Freeze roller coaster from a safe distance of 1000 feet. She notices that she can see the reflection of the highest point of the roller coaster in a puddle of water. Judy is 23.5 feet from that point in the puddle. If Judy is 5feet 3 inches tall, how tall is the roller coaster? #figure(figph[roller coaster], alt: "roller coaster", caption: none) A florist fits a cylindrical piece of foam into a conical vase that is 10 inches high and measures 8 inches across the top, as shown in the figure. If the radius of the foam cylinder is #math.equation(block: false, alt: "2 the fraction 1 over 2")[$2 frac(1, 2)$] inches, how tall should it be just to reach the top of the vase? #figure(figph[cone], alt: "cone", caption: none) #math.equation(block: false, alt: "3 the fraction 3 over 4")[$3 frac(3, 4)$] in To measure the distance across the river shown in the figure, stand at #math.equation(block: false, alt: "A")[$A$] and sight across the river to a convenient landmark at #math.equation(block: false, alt: "B")[$B$]. Then measure the distances #math.equation(block: false, alt: "A C")[$A C$], #math.equation(block: false, alt: "C D")[$C D$], and #math.equation(block: false, alt: "D E")[$D E$]. If #math.equation(block: false, alt: "A C equals 20")[$A C = 20$] feet, #math.equation(block: false, alt: "C D equals 13")[$C D = 13$] feet, and #math.equation(block: false, alt: "D E equals 58")[$D E = 58$] feet, how wide is the river? #figure(figph[river], alt: "river", caption: none) For Problems 41–44, sketch a diagram on graph paper, then solve the problem. Show that the rectangle with vertices #math.equation(block: false, alt: "open parenthesis minus 4 , 1 close parenthesis , open parenthesis 2 , 6 close parenthesis , open parenthesis 7 , 0 close parenthesis")[$( − 4 , 1 ) , ( 2 , 6 ) , ( 7 , 0 )$] and #math.equation(block: false, alt: "open parenthesis 1 , minus 5 close parenthesis")[$( 1 , − 5 )$] is a square. All side have length #math.equation(block: false, alt: "the square root of 61 ,")[$sqrt(61) ,$] opposite sides have slopes #math.equation(block: false, alt: "the fraction 5 over 6")[$display(frac(5, 6))$] and #math.equation(block: false, alt: "the fraction minus 6 over 5")[$display(frac(− 6, 5))$] Show that the points #math.equation(block: false, alt: "open parenthesis 1 , 6 close parenthesis , open parenthesis 5 , 2 close parenthesis , open parenthesis minus 2 , 3 close parenthesis")[$( 1 , 6 ) , ( 5 , 2 ) , ( − 2 , 3 )$] and #math.equation(block: false, alt: "open parenthesis 2 , minus 1 close parenthesis")[$( 2 , − 1 )$] are the vertices of a rectangle. (Hint: If the diagonals of a quadrilateral are of equal length, then the quadrilateral is a rectangle.) Show that the point #math.equation(block: false, alt: "C open parenthesis the square root of 5 , 2 plus the square root of 5 close parenthesis")[$C ( sqrt(5) , 2 + sqrt(5) )$] is the same distance from #math.equation(block: false, alt: "A open parenthesis 2 , 0 close parenthesis")[$A ( 2 , 0 )$] and #math.equation(block: false, alt: "B open parenthesis minus 2 , 4 close parenthesis")[$B ( − 2 , 4 )$]. #math.equation(block: true, alt: "A C equals B C equals 18")[$A C = B C = 18$] Show that the points #math.equation(block: false, alt: "open parenthesis minus 2 , 1 close parenthesis , open parenthesis 0 , minus 1 close parenthesis ,")[$( − 2 , 1 ) , ( 0 , − 1 ) ,$] and #math.equation(block: false, alt: "open parenthesis the square root of 3 minus 1 , the square root of 3 close parenthesis")[$( sqrt(3) − 1 , sqrt(3) )$] are the vertices of an equilateral triangle. + Write an equation that says “The distance from #math.equation(block: false, alt: "open parenthesis x , y close parenthesis")[$( x , y )$] to #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$] is 3 units." + Write an equation for the circle of radius 3 whose center is #math.equation(block: false, alt: "open parenthesis 2 , 5 close parenthesis")[$( 2 , 5 )$]. + #math.equation(block: false, alt: "the square root of open parenthesis x minus 2 close parenthesis squared plus open parenthesis y minus 5 close parenthesis squared equals 3")[$sqrt(( x − 2 )^(2) + ( y − 5 )^(2)) = 3$] + #math.equation(block: false, alt: "open parenthesis x minus 2 close parenthesis squared plus open parenthesis y minus 5 close parenthesis squared equals 9")[$( x − 2 )^(2) + ( y − 5 )^(2) = 9$] The points #math.equation(block: false, alt: "open parenthesis minus 2 , 4 close parenthesis")[$( − 2 , 4 )$] and #math.equation(block: false, alt: "open parenthesis 6 , minus 2 close parenthesis")[$( 6 , − 2 )$] lie on opposite ends of the diameter of a circle. What is the radius of the circle? How long is the diagonal of a rectangle that measures 8 cm by 4 cm? Give an exact value for your answer, and then an approximation rounded to thousandths. #math.equation(block: false, alt: "4 the square root of 5 approximately equals 8.944")[$4 sqrt(5) ≈ 8.944$] cm What is the circumference of a circle of radius 6.2 feet? Give an exact value for your answer, and then an approximation rounded to thousandths. Find two points on the unit circle with #math.equation(block: false, alt: "x")[$x$]-coordinate #math.equation(block: false, alt: "the fraction minus 1 over 3")[$display(frac(− 1, 3))$]. Give exact values for your answers. #math.equation(block: true, alt: "open parenthesis the fraction minus 1 over 3 , the fraction 2 the square root of 2 over 3 close parenthesis , open parenthesis the fraction minus 1 over 3 , the fraction minus 2 the square root of 2 over 3 close parenthesis")[$( display(frac(− 1, 3)) , display(frac(2 sqrt(2), 3)) ) , ( display(frac(− 1, 3)) , display(frac(− 2 sqrt(2), 3)) )$] Find two points on the unit circle with #math.equation(block: false, alt: "y")[$y$]-coordinate #math.equation(block: false, alt: "the fraction the square root of 7 over 4")[$display(frac(sqrt(7), 4))$]. Give exact values for your answers. A circle of radius 10 feet is divided into 5 equal sectors. + Find the arclength of the circular edge of each sector. + Find the area of each sector. + #math.equation(block: false, alt: "4 π")[$4 π$] ft + #math.equation(block: false, alt: "20 π ft squared")[$20 π " " "ft"^(2)$] The central angle of the sector of a circle is #math.equation(block: false, alt: "150 degrees")[$150^(∘)$], and the circle has radius 9 inches. + Find the arclength of the circular edge of each sector. + Find the area of each sector. Delbert slices a 14 inch diameter pizza into 8 equal pieces, and Francine slices a 12 inch diameter pizza into 6 equal slices. Each slice is a sector of a circle. + Find the central angle for the slices. + What are the areas of the slices? Which slices have the greater area? + How long are the crust (curved) edges of the slices? Which slices have the longer crust edges? + #math.equation(block: false, alt: "45 degrees , 60 degrees")[$45^(∘) , 60^(∘)$] + #math.equation(block: false, alt: "the fraction 49 π over 8 in squared , 6 π in squared")[$display(frac(49 π, 8)) " " "in"^(2) , 6 π " " "in"^(2)$] Delbert + #math.equation(block: false, alt: "the fraction 79 π over 4")[$display(frac(79 π, 4))$] in, #math.equation(block: false, alt: "2 π")[$2 π$] in, Francine Florence wants to create a piechart (or circle graph) to display how much of her hospital's budget is dedicated to nurses. She finds that in the hospital's annual expenses of \$60 million, the nurses' salaries and benefits totaled \$1,200,000. + What fraction of the total annual costs comes from the nurses' salaries and benefits? + Suppose that the entire budget is represented by the area of a circle. If the costs for the nurses are to be represented by a sector of that circle, what will be the angle of that sector? + If the circle has a radius of 20 centimeters, what are the areas of the circle and of the sector representing the nurses? What are the circumference of the circle and the arclength of the sector?