1.
An isosceles triangle with vertex angle [latex]100°[/latex]
Chapter 1: Triangles and Circles
Exercises: Chapter 1 Review Problems
Chapter Review Suggested Problems
Problems: #4, 8, 10, 18, 20, 26, 30, 38, 44, 46, 48, 50, 52
Exercises for Chapter 1 Review
Exercise Group
For Problems 1–4, sketch the triangle described.
2.
An isosceles triangle with base angles of [latex]75°[/latex]
3.
A scalene right triangle
4.
A scalene triangle with one obtuse angle
Exercise Group
For Problems 5-16, find the unknown angles.
5.
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16.
Exercise Group
In Problems 17 and 18, name two congruent triangles and find the unknown quantities.
17.
18.
[latex]PQRS[/latex] is a square
Exercise Group
In Problems 19–22, are the pairs of triangles similar? Explain why or why not.
19.
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22.
Exercise Group
In Problems 23–26, find the unknown side.
23.
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26.
Exercise Group
In Problems 27–34, solve for [latex]y[/latex] in terms of [latex]x{.}[/latex]
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34.
Exercise Group
In Problems 35 and 36, find angle [latex]\alpha{.}[/latex] The gray lines are horizontal.
35.
36.
Exercise Group
For Problems 37–40, make a sketch showing similar triangles, write a proportion, and solve.
37.
A [latex]6[/latex]-foot man stands [latex]12[/latex] feet from a lamppost. His shadow is [latex]9[/latex] feet long. How tall is the lamppost?
38.
Judy is observing the Mr. Freeze roller coaster from a safe distance of [latex]1000[/latex] feet. She notices that she can see the reflection of the highest point of the roller coaster in a puddle of water. Judy is [latex]23.5[/latex] feet from that point in the puddle. If Judy is [latex]5[/latex] feet [latex]3[/latex] inches tall, how tall is the roller coaster?
39.
A florist fits a cylindrical piece of foam into a conical vase that is [latex]10[/latex] inches high and measures [latex]8[/latex] inches across the top, as shown in the figure. If the radius of the foam cylinder is [latex]2[/latex] inches, how tall should it be just to reach the top of the vase?
40.
To measure the distance across the river shown in the figure, stand at [latex]A[/latex] and sight across the river to a convenient landmark at [latex]B{.}[/latex] Then measure the distances [latex]AC{,}[/latex] [latex]CD{,}[/latex] and [latex]DE{.}[/latex] If [latex]AC=20[/latex] feet, [latex]CD=13[/latex] feet, and [latex]DE=58[/latex] feet, how wide is the river?
Exercise Group
For Problems 41–44, sketch a diagram on graph paper, then solve the problem.
41.
Show that the rectangle with vertices [latex](-4,1), (2,6), (7,0)[/latex] and [latex](1,-5)[/latex] is a square.
42.
Show that the points [latex](1,6), (5,2), (-2,3)[/latex] and [latex](2,-1)[/latex] are the vertices of a rectangle. (Hint: If the diagonals of a quadrilateral are of equal length, then the quadrilateral is a rectangle.)
43.
Show that the point [latex]C(\sqrt{5},2+\sqrt{5})[/latex] is the same distance from [latex]A(2,0)[/latex] and [latex]B(-2,4){.}[/latex]
44.
Show that the points [latex](-2,1), (0,-1),[/latex] and [latex](\sqrt{3}-1,\sqrt{3})[/latex] are the vertices of an equilateral triangle.
45.
- Write an equation that says “The distance from [latex](x,y)[/latex] to [latex](2,5)[/latex] is [latex]3[/latex] units.”
- Write an equation for the circle of radius [latex]3[/latex] whose center is [latex](2,5){.}[/latex]
46.
The points [latex](-2,4)[/latex] and [latex](6,-2)[/latex] lie on opposite ends of the diameter of a circle. What is the radius of the circle?
47.
How long is the diagonal of a rectangle that measures [latex]8[/latex] cm by [latex]4[/latex] cm? Give an exact value for your answer and then an approximation rounded to thousandths.
48.
What is the circumference of a circle of radius [latex]6.2[/latex] feet? Give an exact value for your answer and then an approximation rounded to thousandths.
49.
Find two points on the unit circle with [latex]x[/latex]-coordinate [latex]\dfrac{-1}{3}{.}[/latex] Give exact values for your answers.
50.
Find two points on the unit circle with [latex]y[/latex]-coordinate [latex]\dfrac{\sqrt{7}}{4}{.}[/latex] Give exact values for your answers.
51.
A circle of radius [latex]10[/latex] feet is divided into [latex]5[/latex] equal sectors.
- Find the arclength of the circular edge of each sector.
- Find the area of each sector.
52.
The central angle of the sector of a circle is [latex]150°{,}[/latex] and the circle has radius 9 inches.
- Find the arclength of the circular edge of each sector.
- Find the area of each sector.
53.
Delbert slices a [latex]14[/latex]-inch-diameter pizza into [latex]8[/latex] 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?
54.
Florence wants to create a pie chart (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 [latex]\$60[/latex] million, the nurses’ salaries and benefits totaled [latex]\$1,200,000.[/latex]
- 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 [latex]20[/latex] 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?