"

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.

1.

An isosceles triangle with vertex angle [latex]100°[/latex]

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.

triangle with 3 sides of 9

6.

right triangle with legs of 21

7.

triangle with angle of 120 between 2 sides of 1

8.

triangle with 3 sides of sqrt 3

9.

right triangle with the hypotenuse extended in both directions; one exterior angle is 115; find its supplementary angle and the other exterior angle

10.

triangle

11.

equilateral triangle with sides of 2 and shown altitude; find a base angle and half of the bisected angle

12.

square with sides of 9 and shown diagonal; find the bisected angle and one of the other angles

13.

triangle inside 2 parallel lines; exterior angle of 141 and interior angle of 62

14.

triangle with exterior angles of 98 and 63

15.

regular heptagon with 7 inscribed triangles; find the central angles and the base angle of a triangle

16.

regular octagon with 8 inscribed triangles; find the central angle and the base angle

Exercise Group

In Problems 17 and 18, name two congruent triangles and find the unknown quantities.

17.

parallel lines

18.

[latex]PQRS[/latex] is a square

square with lines (z) going from the bottom corners to the midpoint (T) of the top side; T is 8 from each of the top 2 corners; find the side of the square and z

Exercise Group

In Problems 19–22, are the pairs of triangles similar? Explain why or why not.

19.

triangles with longest sides of 4 and 5

20.

triangle with sides 6, 7, and 11; and sides 9, 10, 17

21.

triangles

22.

triangles with vertical angles and opposite sides marked similar

Exercise Group

In Problems 23–26, find the unknown side.

23.

isosceles triangle with base 7 and leg 13, find the other leg

24.

isosceles triangle with base 43 and leg 26; find the other leg

25.

triangle with altitude of 24; the base is split into lengths of 32 and t

26.

oblique triangle with an extra base of size 18 drawn parallel to the base of 48; the side of the large triangle is 40 between the 2 parallel lines and s above the side of 18

Exercise Group

In Problems 27–34, solve for [latex]y[/latex] in terms of [latex]x{.}[/latex]

27.

2 triangles formed by intersecting lines and parallel lines for the bases; the longest side is 2+5

28.

triangles drawn by intersecting lines and parallel lines on the bases of 12 and y, the sides of the larger triangle are y, x, and 8

29.

right triangle with an altitude drawn from the right angle; the hypotenuse is labeled as 3+7 on the opposite sides of the altitude

30.

a right triangle has an altitude drawn from the right angle to the hypotenuse of 10; find the base and hypotensue of the smaller triangle

31.

triangles

32.

triangles

33.

a 7-12-13 right triangle is drawn inside a rectangle; creating 4 right triangles

34.

a 55-77-?? right triangle is drawn inscribed in a rectangle creating 4 right triangles

Exercise Group

In Problems 35 and 36, find angle [latex]\alpha{.}[/latex] The gray lines are horizontal.

35.

triangle

36.

triangle

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?

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?

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?

cone

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?

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.
  1. Write an equation that says “The distance from [latex](x,y)[/latex] to [latex](2,5)[/latex] is [latex]3[/latex] units.”
  2. 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.

  1. Find the arclength of the circular edge of each sector.
  2. 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.

  1. Find the arclength of the circular edge of each sector.
  2. 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.

  1. Find the central angle for the slices.
  2. What are the areas of the slices? Which slices have the greater area?
  3. 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]

  1. What fraction of the total annual costs comes from the nurses’ salaries and benefits?
  2. 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?
  3. 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?

License

Icon for the Creative Commons Attribution-ShareAlike 4.0 International License

Trigonometry Copyright © 2024 by LOUIS: The Louisiana Library Network is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.