Extra Practice - Academic Magnet High Schoolamhs.ccsdschools.com/UserFiles/Servers/Server_2856713/File/Renes...Extra Practice Extra Practice ... T R S Identify each transformation

Extra Practice

Extra Practice Skills Practice

Name each of the following.

1. two points 2. two lines

3. two planes 4. a point on � �� IH

5. a line that contains L and J

6. a plane that contains L, K, and H

K

I

H

LJ

Draw and label each of the following.

7. a ray with endpoint A that passes through B

8. a line � �� PQ that intersects plane D

Find each length.

9. MN 10. MO 20 4 6 8 10 12-2-4-6

M N O

11. Segments that have the same length are −−−− ? .

12. Construct a segment congruent to AB. Then construct the midpoint M.

A B

13. M is the midpoint of −−

PR , PM = 2x + 5, and MR = 4x - 7. Solve for x and find PR.

Z is in the interior of ∠WXY. Find each of the following.

14. m∠WXY if ∠WXZ = 23° and m∠ZXY = 51°

15. m∠WXZ if m∠WXY = 44° and m∠ZXY = 20°

�� EH bisects ∠DEF. Find each of the following.

16. m∠DEH if m∠DEH = (10z - 2) ° and m∠HEF = (6z + 10) °

17. m∠DEF if m∠DEH = (9x + 3) ° and m∠HEF = (5x + 11) °

18. A −−−− ? is formed by two opposite rays and measures −−−− ? °.

19. There are −−−− ? ° in a circle.

Tell whether the angles are only adjacent, adjacent and form a linear pair, or not adjacent.

20. ∠AOB and ∠DOE 21. ∠AOE and ∠DOE

22. ∠COE and ∠EOA 23. ∠AOB and ∠BOD

D

CB

A O

E

24. Name a pair of vertical angles.

Given m∠A = 41.7° and m∠B = (24.2 - x) °, find the measure of each of the following.

25. complement of ∠A 26. supplement of ∠A 27. supplement of ∠B

Lesson

1-1

Lesson

1-2

Lesson

1-3

Lesson

1-4

P, V

P

8 14

�

Check students’ constructions.

x = 6, PR = 34

74°

24°

28°

42°

straight ∠; 180

360

20. not adj.21. only adj.22. adj. and a lin. pair23. only adj.Possible answers: ∠AOE and ∠BOC

48.3° 138.3°27. (155.8 + x) °

1–6. Possible answers:

7–8. See Additional Answers.

� ⎯ � LJ

H, I � ⎯ � JK , � ⎯ � IH H

EPS2

Chapter 1

CS10_G_MESE612294_EM_EPSc01.indd EPS2 309011 8:56:32 AM

Find the perimeter and area of each figure.

28.y + 10

12

5y

3y

29.

1.45 cm

30. 2n - 2

3

Find the circumference and area of each circle. Give your answer to the nearest hundredth.

31.

2.6 ft

32.

4 m

33.12.2 in.

34. The formula to find the midpoint M of −−

AB with endpoints A ( x 1 , y 1 ) and B ( x 2 , y 2 ) is −−−− ? .

Find the coordinates of the midpoint of each segment.

35. −−−

WX with endpoints W (-4, 1) and X (2, 9)

36. −−

YZ with midpoints Y (4, 8) and Z (-1, -4)

37. M is the midpoint of −−

RS . R has coordinates (-7, -3) , and M has coordinates (1, 1) . Find the coordinates of S.

Find the length of the given segments and determine if they are congruent.

38. −−−

VW and −−

PQ

39. −−

RS and −−

TU

2

-4

2

y

x

-2

P W

V

Q

U

T

R

S

Identify each transformation. Then use arrow notation to describe the transformation.

40.I�

H� H

G G�

I

41.

A�

B� E�

D� C�

A

B E

D C

42. A figure has vertices at (1, 1) , (2, 4) , and (5, 3) . After a transformation, the image of the figure has vertices at (-3, -2) , (-2, 1) , and (1, 0) . Draw the preimage and image. Then describe the transformation.

43. A figure has vertices at (5, 5) , (2, 6) , (1, 5) , and (2, 4) . After a transformation, the image of the figure has vertices at (5, 5) , (6, 8) , (5, 9) , and (4, 8) . Draw the preimage and image. Then describe the transformation.

44. The coordinates of the vertices of quadrilateral DEFG are (3, 0) , (2, 3) , (-3, 2) , and

(-2, -1) . Find the coordinates for the image of rectangle DEFG after the translation

(x, y) → (x, -y) . Draw the preimage and image. Then describe the transformation.

Lesson

1-5

Lesson

1-6

Lesson

1-7

(-1, 5)

(9, 5)

38. −−

VW = 8 √ � 2 ; −−

PQ = 3 √ � 5 ; no

39. −−

RS = −−

TU = √ � 34 ; yes

rotation; �GHI → �G'H'I'

36. (1 1 _ 2

, 2)



See Additional Answers.

EPS3


Extra Practice Applications Practice

Athletics Use the following information for Exercises 1–3.

During gym class, a teacher notices the following. Decide if each resembles a point, segment, ray, or line. (Lesson 1-1)

1. Kyle starts running in a straight line. Suppose he does not stop running.

2. Agnes runs a quarter-mile in a straight line.

3. Jimmy stands perfectly still.

Travel Use the following information for Exercises 4–6.

The Perez family is driving from Austin, Texas, to Dallas, Texas. The city of Waco is the approximate midpoint between these two cities. It is 102 miles from Austin to Waco. (Lesson 1-2)

4. What is the total distance from Austin to Dallas?

5. The approximate midpoint from Waco to Dallas is Milford. What is the distance from Austin to Milford?

6. The Perez family averages 64 miles per hour. About how long will the entire drive take?

Probability Use the following information for Exercises 7 and 8.

In a carnival game, each contestant spins the wheel and wins the prize indicated by the color. (Lesson 1-3)

Goldfish Balloon

Stuffed Bear

Grand Prize

7. Using a protractor, measure each angle on the wheel.

8. Since there are 360° in a circle, the probability of the wheel landing on a given color is the number of degrees in the angle divided by 360°. Find the probability of the wheel landing on each prize. Express your answer as a fraction in lowest terms.

9. Entomology Because the insect is symmetrical, ∠1 � ∠4 and ∠2 � ∠3. Also, ∠1 and ∠2 are complementary, and ∠3 and ∠4 are complementary. If m∠1 = 48.5°, find m∠2, m∠3, and m∠4. (Lesson 1-4)

4 1 2 3

Architecture Use the following information for Exercises 10 and 11.

The bricks used to make a building are one-fourth as tall as they are wide, and the bricks are 2.25 inches tall. (Lesson 1-5)

10. What is the area of the largest face of each brick?

11. A certain exterior wall is 33 bricks long and 20 bricks tall. What is the area of the wall in square inches?

12. Sports A football coach has his team run sprints diagonally across a football field. If the field is 120 yards long and 160 feet wide, what is the distance they run? Write your answer to the nearest hundredth of a foot. (Lesson 1-6)

13. Crafts The picture below shows half of a stenciled design. The full design should resemble a sun. Name two transformations that can be performed on the image so that the image and its preimage form a complete picture. Be as specific as possible, referring to L and P. (Lesson 1-7)

L

P

ray

seg.pt.

204 mi

153 mi

approximately 3 h 11 min

m∠2 = m∠3 = 41.5°; m∠4 = 48.5°

20.25 in 2

13,365 in 2

393.95 ft

reflection across L, rotation of 180° about P


EPA2

Chapter 1

CS10_G_MESE612294_EM_EPAc01.indd EPA2 309011 7:52:12 AMExtra PracticeEPCH1

CS10_G_METE612331_EM_EPc01.indd 1 4/14/11 11:27:02 PM

Extra Practice


Find the next item in each pattern.

1. 3, 7, 11, 15, … 2. -3, 6, -12, 24, …

3. Complete the conjecture “The product of two negative numbers is −−−− ? .”

4. Show that the conjecture “The quotient of two integers is an integer” is false by finding a counterexample.

Identify the hypothesis and conclusion of each conditional.

5. A number is divisible by 10 if it ends in zero.

6. If the temperature reaches 100° F, it will rain.

Write a conditional statement from each of the following.

7. Perpendicular lines intersect to form 90° angles. 8.

Track

Athletics 9. The sum of two supplementary angles is 180°.

Determine if each conditional is true. If false,give a counterexample.

10. If a figure has four sides, then it is a square.

11. If x = 3 , then 5x = 15 .

12. Does the conclusion use inductive or deductivereasoning? To rent a boat, you must take a boatingsafety course. Jason rented a boat, so Jessicaconcludes that he has taken a boating safety course.

13. Determine if the conjecture is valid by the Law of Detachment.Given: If a student is in tenth grade, then the student may participate in student council. Eric is a tenth-grader.Conjecture: Eric may participate in student council.

14. Determine if the conjecture is valid by the Law of Syllogism.Given: If a triangle is isosceles, then it has two congruent sides. If a triangle has two congruent angles, then it has two congruent sides.Conjecture: If a triangle is isosceles, then it has two congruent angles.

15. Draw a conclusion from the given information.Given: If the sum of the angles of a polygon is 360°, then it is a quadrilateral. If a polygon is a quadrilateral, then it has four sides. The sum of the angles of polygon R is 360°.

16. Write the conditional statement and converse within the biconditional “A triangle is equilateral if and only if it has three congruent sides.”

17. For the conditional “If a triangle is scalene, then its sides have different lengths,” write the converse and a biconditional statement.

18. Determine if the biconditional “n + 3 = -1 ↔ n = -4” is true. If false, give a counterexample.

Write each definition as a biconditional.

19. A parallelogram is a quadrilateral with two pairs of parallel sides.

20. Congruent angles have equal measures.

Lesson

2-1

Lesson

2-2

Lesson

2-3

Lesson

2-4

19 -48

positive

3 ÷ 2 = 1.5

H: A number ends in a zero. C: A number is divisible by 10.

H: The temperature reaches 100° F. C: It will rain.

If 2 lines are ⊥, then they intersect to form 90° �.

If 2 � are supp., then their sum is 180°.

If a student is on the track team, then the student is in athletics.

F; a parallelogram has 4 sides but is not a square.

T

deductive

valid

invalid

Polygon R has 4 sides.

T

A quad. is a parallelogram if and only if it has 2 pairs of parallel sides.

2 � are � if and only if they have = measures.


EPS4

Chapter 2


Solve each equation. Write a justification for each step.

21. 2x + 3 = 9 22. x + 2 _ 5

= 3

Write a justification for each step.

23. AC = AB + BC 9x - 5 = (3x + 6) + (5x + 2)

A B C 3x + 6 5x + 2

9x - 5

9x - 5 = 8x + 8

x - 5 = 8

x = 13

24. Fill in the blanks to complete the two-column proof.

Given: ∠HMK and ∠JML are right angles.

H

J K

L M

2 3

1 Prove: ∠1 � ∠3Proof:

Statements Reasons

1. a. −−−−− ?

2. b. −−−−− ?

c. −−−−− ?

3. ∠1 � ∠3

1. Given

2. Adjacent angles that form a right angle are complementary.

3. d. −−−−− ?

25. Use the given plan to write a two-column proof of the Transitive Property of Congruence.

A

B E

F

C D

Given: −−

AB � −−

CD , −−

CD � −−

EF Prove:

−− AB �

−− EF

Plan: Use the definition of congruentsegments to write the given congruence statements as statements of equality. Then use the Transitive Property of Equality to show that AB = EF. So

−− AB �

−− EF by the

definition of congruent segments.

26. Use the given two-column proof to write a flowchart proof.

Given: ∠2 � ∠3

1 2 3 4 Prove: m∠1 = m∠4Proof:

Statements Reasons

1. ∠2 � ∠3

2. ∠1 and ∠2 are supplementary.

∠3 and ∠4 are supplementary.

3. ∠1 � ∠4

4. m∠1 = m∠4

1. Given

2. Lin. Pair Thm.

3. � Supps. Thm.

4. Def. of � �

27. Use the given two-column proof to write a paragraph proof.

Given: ∠1 � ∠3

1

2

3

4 5

Prove: ∠4 � ∠5Proof:

Statements Reasons

1. ∠1 � ∠3

2. ∠1 � ∠4, ∠3 � ∠5

3. ∠1 � ∠5

4. ∠4 � ∠5

1. Given

2. Vert. � Thm.

3. Trans. Prop. of �

4. Trans. Prop. of �

Lesson

2-5

Lesson

2-6

Lesson

2-7

x + 2 _ 5

= 3 (Given); x + 2 = 15

(Mult. Prop. of =); x = 13 (Subtr. Prop. of =)

Seg. Add. Post.; Subst.; Simplify; Subtr. Prop. of =; Add. Prop. of =

� Comps. Thm.

24a. ∠HMK and ∠JML are rt. �. b. ∠1 and ∠2 are comp. c. ∠2 and ∠3 are comp.



It is given that ∠1 � ∠3. By the Vert. � Thm., ∠1 � ∠4 and ∠3 � ∠5. By the Trans. Prop. of �, ∠1 � ∠5. Similarly, ∠4 � ∠5.

EPS5



1. Health Mike collected the following data about the heights of twelve students in his tenth-grade class. Use the table to make a conjecture about the heights of boys and girls in the tenth grade. (Lesson 2-1)

Height (in.) of Tenth-Grade Students

Boys 70 71 68 67 70 67

Girls 67 64 64 65 68 66

2. Government Voter Turnout

Year Voters

1996 12,530

1998 8,750

2000 15,210

2002 7,370

2004 14,380

Presidential elections are held every four years. Elections for senators are held every two years. So in years not divisible by 4, only Senate seats are up for election. The table shows voter turnout for a small town during recent election years. Make a conjecture based on the data. (Lesson 2-1)

3. Biology Write the converse, inverse, and contrapositive of the conditional statement “If an animal is a fish, then it swims in salt water.” Find the truth value of each. (Lesson 2-2)

4. Gardening Write the converse, inverse, and contrapositive of the conditional statement “If a plant is watered, then it will grow.” Find the truth value of each. (Lesson 2-2)

5. Sports Determine if the conjecture is valid by the Law of Detachment. (Lesson 2-3)

Given: If you participate in a triathlon, then you run, swim, and bike. Margie runs, swims, and bikes.

Conjecture: Margie participates in a triathlon.

6. Health Students are required to have certain immunizations before attending school to prevent the spread of disease. Write the conditional statement and converse within the biconditional “Students can attend public school if and only if they have the required immunizations.” (Lesson 2-4)

7. Weather Hurricanes are assigned category numbers to describe the amount of flooding and wind damage they are likely to cause. Write the statement “If a hurricane has sustained winds of more than 155 miles per hour, then it is Category 5” as a biconditional statement. (Lesson 2-4)

8. Athletics The equation c = 5w + 25 relates the number of workouts w to the cost c of a weight training group. If Matthew plans to spend $200 on weight training, how many workouts can he participate in? Solve the equation for w and justify each step.(Lesson 2-5)

9. Nutrition Rick has allotted himself 200 Calories for his evening snack, which consists of a glass of milk and crackers. A glass of milk has 110 Calories, and each cracker has 15 Calories. The equation s = 110 + 15c relates the number of crackers c to the total number of Calories s in Rick’s evening snack. How many crackers can Rick have? Solve the equation for c and justify each step.(Lesson 2-5)

10. Travel On a city map, the library, post office, and police station are collinear points in that order. The distance from the library to the post office is 2.3 miles. The distance from the post office to the police station is 5.1 miles. Which theorem can you use to conclude that the distance from the library to the police station is 7.4 miles? (Lesson 2-6)

11. Recreation Kyle is making a kite from the pattern below by cutting four triangles from different pieces of material. Write a paragraph proof to show that m∠3 = 90°. (Lesson 2-7)

2 1

3 4

Given: ∠1 � ∠2Prove: m∠3 = 90°

invalid

Seg. Add. Post.









11. It is given that ∠1 � ∠2. By the Lin. Pair Thm., ∠1 and ∠2 are supplementary. By Thm. 2-7-3, ∠1 and ∠2 are rt. �. So m∠1 = 90°. By the Vert. ∠s Thm., ∠1 � ∠3, so m∠1 = m∠3 by the def. of � �. By subst., m∠3 = 90°.

EPA3

Chapter 2



Extra Practice


Identify each of the following.

1. a pair of parallel segments

J K

M N

Q P

H L

2. a pair of perpendicular segments

3. a pair of skew segments

Identify the transversal and classify each angle pair.

4. ∠5 and ∠3

1 2 4 3

5

p q

r

5. ∠2 and ∠4

6. ∠5 and ∠1

Find each angle measure.

7. m∠XYZ

(8x - 15)˚

(3x + 20)˚ X

Y

Z

8. m∠KJH x˚

125˚ H

J

K

9. m∠ABC

C

B

A (21x)˚

(13x + 24)˚

10. m∠LMN

(21x + 14)˚

(18x + 10)˚

M N

L

11. m∠PQR Q

R

P

15x˚

(12x + 17)˚

12. m∠TUV T U

V (3x + 18)˚

(9x - 6)˚

Use the Converse of the Corresponding Angles Postulate and the given information to show that � � m.

13. ∠2 � ∠4

�

m

2 1 3 8

4 7 5 6

14. m∠8 = 5x + 36, m∠6 = 11x + 12, x = 4

Use the theorems and given information to show that p � q.

15. ∠1 � ∠8 2 1 3

8

4

7 5 6 �

qp

16. m∠2 = 9x + 31, m∠3 = 6x + 14, x = 9

17. Write a two-column proof.

Given: ∠1 and ∠5 are supplementary.

2 1

4

8

3

7

6 5

�

m Prove: � � m

Lesson

3-1

Lesson

3-2

Lesson

3-3

−−

JK and −−

HL

−−

JK and −−

HJ

Possible answer: −−

JK and −−

HQ

q, alt. int.

r, corr.

p, alt. ext.

41°

63°

85°

125°

98°

120°

∠2 � ∠4, so � � m by the Conv. of Corr. � Post.

∠1 � ∠8, so p � q by the Conv. of Alt. Ext. � Thm.



EPS6

Chapter 3


18. Name the shortest segment from point A to � �� BE .

19. Write and solve an inequality for x. 9

B C D

A

E

3x + 5

Solve for x and y in each diagram.

20.

(7x - 2y)˚

(5x)˚

21. (3x + 4y)˚

(9x)˚


Given: � ⊥ p, m ⊥ p

� m

p Prove: � � m

Use the slope formula to determine the slope of each line.

23. �� FG

2 4

-2

4

2

y

x

F

G

-2

24. � �� HJ

2 4

-2

2

y

x

-2 0

5

H J

Graph each pair of lines. Use slopes to determine whether the lines are parallel, perpendicular, or neither.

25. �� AB and � �� CD for A (4, 7) , B (3, 2) , C (-3, 4) , D (2, 3)

26. �� EF and � �� GH for E (-2, 4) , F (3, 1) , G (-1, -2) , H (4, -5)

27. �� JK and � �� LM for J (-3, 3) , K (4, -2) , L (4, 2) , M (0, -4)

Write the equation of each line in the given form.

28. the line with slope - 2 _ 3

through (3, -1) in point-slope form

29. the line through (-2, 2) and (4, -1) in slope-intercept form

30. the line with x-intercept -3 and y-intercept 4 in slope-intercept form

Graph each line.

31. y = - 3 _ 4

x + 2 32. y + 4 = -3 (x + 2)

33. y = 2 34. x = -1

Determine whether the lines are parallel, intersect, or coincide.

35. y = 4x + 2, 4x - y = 1 36. y =- 1 _ 2

x + 3, 2y + x = 6

37. 2x + 5y = 1, 5x + 2y = 1 38. 2x - y = 5, 2x - y = -5

Lesson

3-4

Lesson

3-5

Lesson

3-6

−−

AC

3x + 5 > 9, x >

4 _ 3

x = 18, y = 18

x = 10, y = 15

1 0

25–27. For graphs, see Additional Answers.

perpendicular

parallel

neither


y = 4 _ 3

x + 4

y = -

1 _ 2

x + 1

y + 1 = -

2 _ 3

(x - 3)


parallel

intersect

coincide

intersect

EPS7



1. Recreation A scuba diver leaves a flag on the surface of the water to alert boaters of his location. Describe two parallel lines and a transversal in the flag. (Lesson 3-1)

2. Carpentry In the stairs shown, the horizontal treads and the vertical risers are all parallel. m∠1 = (14x + 6) ° and m∠2 = (19x - 24) °. Find x. (Lesson 3-2)

1

2

3. Transportation The train tracks shown cross the street lanes. The lanes of the street are parallel. Find x in the diagram. (Lesson 3 -3)

125˚

(3x + 25)˚

4. Sports At a track meet, the starting blocks are placed along a line that is a transversal to the lanes. m∠1 = 12x - 8, m∠2 = 8x + 12, and x = 5. Show that the lines between the lanes are parallel. (Lesson 3-3)

1

2

3

5. Transportation The railroad ties in the diagram are all parallel. m∠1 = 19x - 5 and m∠ 2 = 4x + 5y. Find x and y so that the ties are all perpendicular to the tracks. (Lesson 3-4)

2 1

6. Art The sides of a picture frame are cut so that the opposite sides of the frame are parallel and the consecutive sides are perpendicular. Find the values of x and y in the diagram. (Lesson 3-4)

(13x + y)˚ (3x + 3y)˚

7. Recreation At 1:00 P.M., a boat on a river passes a point that is 3 miles from a lodge. At 5:30 P.M., the boat passes a point that is 8 miles from the lodge. Graph the line that represents the boat’s distance from the lodge. Find and interpret the slope of the line. (Lesson 3-5)

8. Sports A marathon runner runs 10 miles by 3:00 P.M. and 25 miles by 4:30 P.M. Graph the line that represents her distance run. Find and interpret the slope of the line. (Lesson 3-5)

9. Business A cab company charges $8 per ride plus $0.25 per mile. Another cab company charges $5 per ride plus $0.35 per mile. For how many miles will two cab rides cost the same amount? (Lesson 3-6)

10. Food A pizza parlor is catering a schoolevent. Pete’s Pizza charges $85 for the first20 students and $5 for each additional student. Polly’s Pizza charges $125 for the first20 students and $3 for each additional student. For how many students will the pizza parlors cost the same? (Lesson 3-6)

Possible answer: The top and the bottom of the flag are �. The white stripe is a transv.

x = 6

x = 10

x = 5, y = 14

x = 5, y = 25

30 miles

40 students




EPA4

Chapter 3



Extra Practice


Apply the transformation M to the polygon with the given vertices. Name the coordinates of the image points. Identify and describe the transformation.

1. M: (x, y) � (–x, y) 2. M: (x, y) � (x – 2, y + 2)

A(4, 5), B(1, 3), C(2, 6) P(–3, 2), Q(5, 0), R(4, –2)

Determine whether the polygons with the given vertices are congruent. Support your answer by describing a transformation.

3. X(1, –3), Y(0, 2), Z(–1, 4) and M(3, –9), N(0, 6), P(–3, 12)

Classify each triangle by its angle measures.

4. �ABC

A C D

B

60˚ 60˚

30˚ 30˚

5. �BCD

Classify each triangle by its side lengths.

6. �EFG F

E H G

7. �FGH

8. �EFH

9. Find the side lengths of �JKL. K L

2x + 6

2x + 5

3x - 8

J

The measure of one of the acute angles of a right triangle is given. What is the measure of the other acute angle?

10. 38° 11. 27.6°


12. m∠A (10x - 5)˚

(8x + 3)˚ 124˚

A

B C D

13. m∠J and m∠P

N K

L J

M P

(2x2 - 25)˚ (x2)˚

Given: �GHI � �JKL. Identify the congruent corresponding parts.

14. −−−

GH � −−−− ? 15. −−

JL � −−−− ? 16. ∠K � −−−− ?

Given: �LMN � �PQN. Find each value.

17. x N

Q

M

P

L (8x + 12)˚

(10x - 3)˚

18. m∠LMN

Use SSS to explain why the triangles in each pair are congruent.

19. �QRS � �QRT 20. �UVW � �WXU

X W

U V

R T S

Q

Show that the triangles are congruent for the given value of the variable.

21. �XYZ � �ABC, 22. �DEF � �GFE, x = 4 y = 8X A

Y B

Z C

2x - 3 5x - 8

6x - 9

5

15

12

D

E

G

F

5y - 3

3y + 13

(12y - 6)˚

Lesson

4-1

Lesson

4-2

Lesson

4-3

Lesson

4-4

Lesson

4-5

equiangular

rt.

equil.

isosc.

scalene

JL = 31, KL = 31, JK = 32

52° 62.4°

65° 25°

−−

JK −−

GI ∠H

7.5

18°

19–22. See Additional Answers

3. No, the triangles are not congruent because triangle XYZ can be mapped to triangle

MNP by a dilation with a scale factor of 3 and a center of (0, 0).

2. P(–5, 4), Q(3, 2), R(2, 0)This is a translation 2 units left and 2 units up.

1. A'(–4, 5), B'(–1, 3), C'(–2, 6) This is a reflection across the y-axis.

EPS8

Chapter 4


Determine if you can use ASA to prove the triangles congruent. Explain.

23. �ACB and �ACD 24. �EFG and �HGF F E

G H C

B D

A

Determine if you can use the HL Congruence Theorem to prove the triangles congruent. If not, tell what else you need to know.

25. �ABC � �EDC 26. �FGH � �FJHB

C

D E

A

F

G J H

27. Given: −−−

MN � −−

LP , 28. Given: ∠1 � ∠6, ∠4 � ∠6∠N � ∠L ∠1 � ∠3,

−− AB �

−− AE

Prove: −−−

ML � −−

PN Prove: �ACD is isosceles.

29. Given: �ABC with vertices A (2, 4) , B (3, 1) , C (5, 2) and �DEF with vertices D (-4, -2) , E (-1, -3) , F (-2, -5)

Prove: ∠BAC � ∠EDF

Position each figure in the coordinate plane.

30. a rectangle with length 7 units and width 3 units 31. a square with side length 3a

Write a coordinate proof.

32. Given: Right �GHI has coordinates G (0, 0) , H (0, 4) , and I (6, 0) . J is the midpoint of

−−− GH , and K is the midpoint of

−− GI .

Prove: The area of �GJK is 1 __ 4 the area of �GHI.

Assign coordinates to each vertex and write a coordinate proof.

33. Given: A is the midpoint of −−−

XW in rectangle WXYZ. B is the midpoint of

−− YZ .

Prove: AB = XY


34. m∠X 35. m∠A B

A C (3x + 5)˚ (6x - 10)˚

Find each value.

36. x

(5x + 10)˚

37. y

9y - 3

7y + 5

38. Given: �XYZ is isosceles.A is the midpoint of

−− XZ .

−− XY �

−− YZ

Prove: �YAZ is isosceles.

Lesson

4-6

Lesson

4-7

Lesson

4-8

Lesson

4-9

A

B E

C D

2 5

1 6 3 4

J

G

y

x

2 -2 4 0

2

-2

H

K

I

M N

P L

X

Y

Z 52˚

A

X

Z Y

y

x

64° 20°

10 4


See Additional Answers

EPS9



1. Crafts On a coordinate plane, patterns for two pieces of stained glass have coordinates A(4, 1), B(3, 5), C(1, 2) and D(1, –3), E(5, –2), F(2, 0). Prove that the patterns are congruent. (Lesson 4-1)

2. Camping Three poles are used to create the frame for a tent. The front of the tent is an isosceles triangle with

−− AB �

−− BC . The length of

the base is 1.5 times the length of the sides. The perimeter of the triangle is 21 ft. Find each side length. (Lesson 4-2)

3. Geography The universities in Durham, Chapel Hill, and Raleigh, North Carolina form what is known as the Research Triangle. Use the map to find the measure of the angle whose vertex is at Durham. (Lesson 4-3)

Durham

Chapel Hill

Raleigh

29º 63º

4. Business Oil derricks are used as supports for oil drilling equipment. Use the diagram to prove the following. (Lesson 4-4)

Given: −−

AB � −−−

HG , −−

HB � −−

AG ∠GAB � ∠BHG, ∠AGB � ∠HBG

Prove: �AGB � �HBG

5. Sports A kite is made up of two pairs of congruent triangles. Use SAS to explain why �ABD � �CBD. (Lesson 4-5)

D B E

A

C

6. Recreation A student is estimating the height of a water slide. From a certain distance, the angle from where he is standing to a point on the highest part of the slide is 35°. From a distance 200 m closer, the same angle is 45°. Which postulate or theorem can be used to show that the triangle with the point at the top of the slide as one vertex, and the points where the measurements were taken as the other vertices, is uniquely determined? (Lesson 4-6)

7. Surveying To find the distance AB across a lake, first locate point C. Then measure the distance from C to B. Locate point D the same distance from C as B, but in the opposite direction. Then measure the distance from C to A and locate point E in a similar manner. What is the distance AB across the lake? (Lesson 4-7)

D

E

C

B 120 yd

135 yd

97 yd

A

8. The first step in creating a Sierpinski triangle is to connect the midpoints of the sides of a triangle as shown. (Lesson 4-8)

A

D E

B C F

Given: Equilateral �ABC, D is the midpoint of

−− AB , E is the midpoint of

−− AC , and F is

the midpoint of −−

BC .

Prove: The area of �DEF is 1 __ 4 the area of �ABC.

9. Recreation A boat is sailing parallel to the coastline along � �� XY . When the boat is at X, the measure of the angle from the lighthouse W to the boat is 30°. After the boat has traveled 5 miles to Y, the angle from the lighthouse to the boat is 60°. How can you find WY? (Lesson 4-9)

A

B C

H

G

60˚ 5 mi.

W

Y

X Z

30˚

AB = 6, BC = 6, AC = 9

88°

135 yd

9. Since the boat is parallel to the coastline,

−−

WZ � −−

YX . So ∠ZWX � ∠WXY. Since m∠YWZ = 60° and m∠XWZ = 30°, by subtr., m∠YWX =

30°. Therefore �WYX is isosc. and

−−

WY � −−

YX . Thus WY = 5.





AAS or ASAEPA5

Chapter 4



Extra Practice


Find each measure.

1. CD 8

B

A

D

C

2. HG

9.4

H

E F G 3. JM J

L M K

4x - 10 3x + 5

4. m∠SRT, given m∠SRU = 126° 5. PQ 6. m∠WXV

36 36

R

T

S U

21.5

N

P

O Q

(6x - 6)˚ (4x + 8)˚

W

Y

V X

7. Write an equation in point-slope form for the perpendicular bisector of the segment with endpoints A (1, 4) and B (-5, -2) .

−−

DG , −−

EG , and −−

FG are the perpendicular bisectors of �ABC. Find each length.

8. BG 9. AG 3.9

3.5

3.8

3.02

E

C F

G B

D

A

Find the circumcenter of a triangle with the given vertices.

10. H (5, 0) , J (0, 3) , K (0, 0) 11. L (0, 0) , M (-2, 0) , N (0, -4)

−−

QS and −−

RS are angle bisectors of �QPR. Find each measure.

12. the distance from S to −−

PR 13. m∠SQP

Q R

P

S

72˚

27˚ 6.4

In �DEF, DJ = 30, and FM = 12. Find each length.

14. DM 15. MJ

16. GF 17. GM M

H

G

F

J

E

D

Find the orthocenter of a triangle with the given vertices.

18. N (-2, 2) , P (4, 2) , Q (0, -2) 19. R (-2, 1) , S (2, 5) , T (4, 1)

20. The vertices of �WXY are W (-3, 2) , X (5, 2) , and Y (1, -4) . A is the midpoint of −−−

WY ,

and B is the midpoint of −−

XY . Show that −−

AB � −−−

WX and AB = 1 __ 2 WX.

Find each measure.

21. DE 22. FG

23. DG 24. m∠CHF

25. m∠FHE 26. m∠CED 47˚

10.4

E G

F H

C

D

6.5

Lesson

5-1

Lesson

5-2

Lesson

5-3

Lesson

5-4

8 9.4 50

21.5 36°63°

5.4 5.4

(2 1 _ 2

, 1 1 _ 2

) (-1, -2)

6.4 27°

20 10

18 6

(0, 0) (2, 3)

13 5.2

6.5 47°

133° 47°

y – 1 = –1 (x + 2)


EPS10

Chapter 5


Write an indirect proof of each statement.

27. An isosceles triangle cannot have an obtuse base angle.

28. A right triangle cannot have three congruent sides.

29. Write the angles in 30. Write the sides in order from smallest order from shortest

to largest. 21 37.5

40 J

H

K to longest.

47˚ M L

N

Tell whether a triangle can have sides with the given lengths. Explain.

31. 4, 7, 8 32. 7, 9, 18 33. 2x + 5, 4x, 3 x 2 , when x = 3

The lengths of two sides of a triangle are given. Find the range of possible lengths for the third side.

34. 4 in., 10 in. 35. 8 ft, 8 ft 36. 6.2 cm, 12 cm

Compare the given measures.

37. Compare RS and UV. 38. Compare m∠XWY 39. Find the range of and m∠ZWY. values for x.

5 5 6 6

T

U V S R

Q

78˚ 85˚ 4

5

10

8

8

Y

Z

X

W

6x + 5

13

30˚

28.5˚


Given: m∠X > m∠Y, m∠B > m∠A Prove: AY > XB

Z

A

B

X

Y Find the value of x. Give your answer in simplest radical form.

41.

x

7 10 42. x

12 18

43. x x - 3

6

Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain.

44. 30

40

45.

6 9

46. 7

25

Tell if the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.

47. 4, 7.5, 8.5 48. 6, 10, 11 49. 9, 21, 25

Find the value of x. Give your answer in simplest radical form.

50. x

8

45˚

51.

x

10

52. x

3 √ � 2 45˚

Find the values of x and y. Give your answers in simplest radical form.

53. 30˚

x

y 9

54.

60˚ x y

12 55. 60˚ 30˚

x

y 6 √ � 3

Lesson

5-5

Lesson

5-6

Lesson

5-7

Lesson

5-8

∠K, ∠J, ∠H −−

LN , −−

MN , −−

LM



RS < UV

m∠XWY > m∠ZWY

- 5 _ 6

< x < 4 _ 3

7.5

√ �� 149 6 √ � 5

50; yes

3 √ � 13 ; no 24; yes

yes; right yes; acute yes; obtuse

8 √ � 2 5 √ � 2 3


9 √ � 3 ; 18 4 √ � 3 ; 8 √ � 3 12 √ � 3 ; 18

EPS11


D

FE50 m

H 75 m G90 m

5 ft

100 ft

60º

Seesaw

Swings

Slide

House

400 ft

250 ft School

Park


1. Building The guy wires −−

AB and −−

CB supporting a cell phone tower are congruent and are equally spaced from the base of the tower. How do these wires ensure that the cell phone tower is perpendicular to the ground? (Lesson 5-1)

B

CD

A

2. Safety City planners want to relocate their town’s firehouse so that it is the same distance from the three main streets of the town. Draw a sketch to show where the firehouse should be positioned. Justify your sketch.(Lesson 5-2)

3. Safety A lifeguard needs to watch three areas of a water park. Draw a sketch to show where she should stand to be the same distance from all the swimmers. Justify your sketch.(Lesson 5-2)

4. Art An artist is designing a sculpture composed of a pedestal with a triangular top. The vertices of the top are A (-4, 2) , B (2, 4) , and C (4, -3) . Where should the artist attach the pedestal so that the triangle is balanced? (Lesson 5-3)

5. Measurement City engineers plan to build a bridge across the pond shown. What will be the length of the bridge, GH? (Lesson 5-4)

Engineering Use the following information for Exercises 6 and 7.

Playground engineers are planning a sidewalk that will connect the swings, seesaw, and slide.(Lesson 5-5)

6. If the angle at the swings is the largest, which portion of the sidewalk will be the longest?

7. The distance from the swings to the slide is 37 ft. Can the lengths of the other sides be 40 ft and 50 ft? Explain.

8. Geography The cities of Allenville, Baytown, College City, and Dean Park are shown on the map. Baytown and Dean Park are each 30 miles from College City. Which city is closer to Allenville: Baytown or Dean Park? (Lesson 5-6)

30 mi

30 mi

Allenville

Baytown

College City

Dean Park

47˚ 53˚

9. Mark is late for school. He usually goes around the park so he can walk along the water. Today he decides to cut through the park. About how many feet does he save by going through the park? (Lesson 5-7)

10. Sports A baseball diamond is a square with a side length of 90 ft. What is the distance from first base to third base? (Lesson 5-8)

11. Recreation Haley, who is 5 ft tall, is flying a kite on 100 ft of string. How high is the kite? (Lesson 5-8)

Central Ave.

Main St.

Oak St.

Splash park

Children’s area

Lap pool

25 m

Baytown

about 178 ft

5 + 50 √

�

3 ft, or about 91.6 ft







EPA6

Chapter 5



Extra Practice


Tell whether each figure is a polygon. If it is a polygon, name it by the number of its sides.

1. 2. 3.

Tell whether each polygon is regular or irregular. Tell whether it is concave or convex.

4. 5. 6.

7. Find the measure of each interior angle of pentagon ABCDE.

3x˚A

B

CD

E

2x˚

(2x - 15)˚

(x + 20)˚

(2x + 5)˚ 8. Find the sum of the interior angle measures of a convex heptagon.

9. Find the measure of each interior angle of a regular 15-gon.

10. Find the value of x in polygon FGHJKL.

3x˚

4x˚

4x˚

5x˚

5x˚ 3x˚

FG

J

H

K

L

11. Find the measure of each exterior angle of a regular dodecagon.

MNOP is a parallelogram. Find each measure.

12. MP 13. m∠M 14. m∠N

M N

OP

(13x + 12)˚3x + 8 6x - 10

(21x - 2)˚

Three vertices of �QRST are given. Find the coordinates of T.

15. Q (-5, 3) , R (3, 6) , S (6, 4) 16. Q (-1, 7) , R (3, 3) , S (-2, 3)

Write a two-column proof.

17. Given: ABFG and HDEG are parallelograms.

Prove: ∠B � ∠D

C

B

F E

D

G

H

A

18. Show that RSTU is a parallelogram 19. Show that WXYZ is a parallelogramfor x = 2 and y = 3. for a = 6 and b = 11.

S

TU

R

4x + 3

8y + 3

5y + 12

9x - 7

W

X

Z

Y(13a + 8)˚(8b + 6)˚

(15a - 4)˚ Determine if each quadrilateral must be a parallelogram. Justify your answer.

20. 21. 22.

Show that the quadrilateral with the given vertices is a parallelogram.

23. W (0, 0) , X (-3, 3) , Y (5, 5) , Z (8, 2) 24. A (-3, 1) , B (-2, 4) , C (1, 2) , D (0, -1)

Lesson

6-1

Lesson

6-2

Lesson

6-3

polygon, hexagon

not a polygon polygon, dodecagon

irregular, concave irregular, concave

regular, convex

900°

156°

15

30°

26 77° 103°

(-2, 1)


(-6, 7)



EPS12

Chapter 6


EFGH is a rectangle. Find each measure.

25. EH 26. HF

3y +

407y

- 10

E F

GH

D10x - 8 9x + 5

JKLM is a rhombus. Find each measure.

27. JK 28. m∠NKL

(5y + 5)˚

(8y + 7)˚

6x - 5 4x + 3

N

K

LJ

M Show that the diagonals of a square with the given vertices are congruent perpendicular bisectors of each other.

29. N (1, 4) , P (4, 1) , Q (1, -2) , R (-2, 1) 30. S (-2, 7) , T (2, 8) , U (3, 4) , V (-1, 3)

31. Given: WXYZ is a rectangle. −−

XB � −−

AZ Prove:

−−− WB �

−− YA

W

X B

ZA

Y

Determine if the conclusion is valid. If not, tell what additional information is needed to make it valid.

32. Given: −−

XY � −−−

WZ , −−

XY � −−−

WZ , −−

XZ ⊥ −−−

WY Conclusion: WXYZ is a rhombus.

33. Given: −−−

WX � −−

XY Conclusion: WXYZ is a square.

X Y

ZW 34. Given: −−−

WX ⊥ −−

XY , −−−

WX ⊥ −−−

WZ Conclusion: WXYZ is a rectangle.

Use the diagonals to determine whether a parallelogram with the given vertices is a rectangle, rhombus, or square. Give all the names that apply.

35. A (1, 0) , B (2, -4) , C (6, -3) , D (5, 1) 36. E (-3, -1) , F (-4, -4) , G (2, -6) , H (3, -3)

In kite TUVW, m∠XTU = 65°, and m∠UVT = 32°. Find each measure.

37. m∠TUX 38. m∠XUV 39. m∠TWX

V

U

XT

W

Find each measure.

40. m∠C 41. HJ, given that EG = 32.8

A

D C

B125˚

and FJ = 24.3

H E

J

G F

42. Find the value of x so 43. Given RP = 8y - 7 and NQ = 10y - 12, that JKLM is isosceles. find the value of y so that NPQR is isosceles.

LM

KJ

(4x² + 2x -1)˚

(3x² + 2x + 8)˚

RN

QP

44. Find RS. 45. Find XY.

SR

W T

UV

25

41

A

X Y

D C

B

26 cm

18 cm

Lesson

6-4

Lesson

6-5

Lesson

6-6

122 155

19 35°

square, rect., rhombus rect.

25° 58° 25°

55°

8.5

±32.5

valid



922 cm

EPS13



1. Safety A stop sign is in the shape of a regular octagon. What is the value of x? (Lesson 6-1)

STOPx˚

2. Hobbies Nancy is planting a garden shaped like a regular pentagon. She bought metal edging to surround the garden and prevent weeds. What angle should the edging form at the vertices of the garden? (Lesson 6-1)

x˚

Fishing Use the following information for Exercises 3–5.

The hinges for the trays in a tackle box form parallelograms to ensure that the trays stayparallel to the base of the box. In �ABCD,AB = 21 in., AE = 9 in., and m∠BCD = 125°.Find each measure. (Lesson 6-2)

A B

CD

E

3. DC

4. EC

5. m∠ADC

6. Design A glide rocker uses hinged parallelograms to move the chair back and forth. In �ABCD, AB = DC, and AD = BC. The sides of the parallelogram,

−− AD and

−− BC ,

rotate together to move the chair. Why is ABCD always a parallelogram? (Lesson 6-3)

Design Use the following information for Exercises 7–9.

When extended, the legs of a folding table must form a rectangle so the tabletop is parallel to the ground. Given that JK = 48 in. and KN = 36 in., find each length. (Lesson 6-4)

7. JM

8. JN

9. NM

10. Hobbies Elise is creating a decorative page for her scrapbook. She has a piece of ribbon that is 12 inches long. She wants to outline a rhombus with the ribbon. How can Elise cut the ribbon to ensure that the final shape is a rhombus? (Lesson 6-5)

11. Carpentry Luke is cutting a rectangular window frame. The dimensions of the window are to be 3 feet by 4 feet. What should the diagonal of the frame measure so that the window is rectangular? (Lesson 6-5)

12. Hobbies Addie is making a kite with diagonals of 32 inches and 18 inches. She wants to put a ribbon around the edge of the kite. She will add an 8-foot tail to the kite, made of the same ribbon. If ribbon can be purchased in packages of 3 yards, how many packages should she buy for the entire project? (Lesson 6-6)

9 in.9 in.

12 in.

20 in.

13. Carpentry Aaron is building a shadow box for his baseball memorabilia. The shadow box will be in the shape of a trapezoid, as shown below. The wood for the box costs $1.59 per foot. Estimate the cost of the lumber.(Lesson 6-6)

60˚30 in.

10 in. 10 in.

36 in.

48 in.

45°

108°

21 in.

9 in.

55°

36 in.

60 in.

48 in.

5 ft

2 packages

about $9.28



EPA7

Chapter 6



Extra Practice


Identify the pairs of congruent angles and corresponding sides.

1.

B C

8

7

12

A

E F

6 4

3.5

D

2. 45

36 N M

L K

21 21

12 7 7

I J

G H

15

Determine whether the polygons are similar. If so, write the similarity ratio and a similarity statement.

3. rectangles ABCD and EFGH 4. �JKL and �MNO

A

E F

G H

D C

B 6 2

3

9

L

J M

K O N

8 5

2 3

6 4

Name the coordinates of the image points. Identify and describe the transformation.

5. D : (x, y) → (2x, 2y) 6. D : (x, y) → (1.5x, 1.5y)

A(–2, –1), B(2, 3), C(0, 4) M(–6, 2), N(4, 0), P(5, –8)

7. D : (x, y) → (0.5x, 0.5y) 8. D : (x, y) →(4x, 4y)

R(–2, 0), S(10, –7), T(–2, –6) E(–0.5, 0.5), F(1, –2.5), G(–2, 0.75)

Determine whether the polygons with the given vertices are similar. Support your answer by describing a transformation.

9. E(4, 0), F(–4, –12), G(8, –8) 10. P(–1, 4), Q(3, 2), R(1, –2)

J(3, 0), K(–3, –9), L(6, –6) X(4, 4), Y(12, 0), Z(8, –8)

Explain why the triangles are similar and write a similarity statement.

11. A B

C D

12.

R

U T

S

Q Verify that the triangles are similar.

13. �FGH ∼ �JKH 14. �ACE ∼ �BCD

8 7.5

8.5

17 15

4

F

J G

K

H

6

12 10

5

30 20

C

D

E A

B

Explain why the triangles are similar and then find each length.

15. �XYZ and �ABC, BC 16. �RSV and �UST, TU

X

Z

Y A B

C

18

15 9

6

3

SR

V U

T

5

10 9

Lesson

7-1

Lesson

7-2

Lesson

7-3

The � are �.

(x, y) → ( 3 _ 4

x, 3 _ 4

y) .

∠A � ∠D, ∠B � ∠E, ∠C � ∠F; AB _ DE

= AC _ DF

= BC _ EF

12. Possible answer: 1 _ 3

; ABCD ∼ EFGH



5. dilation about (0, 0) with a scale

9. The polygons are similar; EFG



to �P �Q �R � by a translation: (x, y) →

10. The polygons are similar; �PQR can be mapped

8. dilation about (0, 0) with a scale factor

factor of 2; A�(–4, –2), B�(4, 6), C�(0, 8)

can be mapped to JKL by a dilation:

factor of 0.5; R�(–1, 0), S�(5, –3.5), T �(–1, –3)

factor of 1.5; M �(–9, 3), N�(6, 0), P �(7.5, –12)

(x + 3, y – 2). Then �P �Q �R � can be mapped to �XYZ by a dilation: (x, y) → (2x, 2y).

of 4; E �(–2, 2), F �(4, –10), G �(–8, 3)

EPS14

Chapter 7


Find the length of each segment.

17. −−

AE 18. −−

KJ

8 10

3 A B

C E

D

7

5 6 F

H G

J

K

Verify that the given segments are parallel.

19. −−

EF and −−

JG 20. −−

LP and −−−

MN

3 15

20

4

E

H G

J

F

16 12

15 20

K

P

N M

L

Find the length of each segment.

21. −−

RS and −−

ST 22. −−−

XW and −−−

WZ

10 12

3x - 3 x + 5 S T R

Q

W

Y X

2x + 3 7x - 12

Z

9

18

The scale drawing of the playhouse is 1 in. : 10 ft. Find the actual lengths of the following walls.

23. −−−

GH

24. −−

EF 0.5 in.1 in.

4 in.

1.5 in.2 in.

DA

B

G

H

E

C

KJF 25.

−− DC

The school courtyard is 25 ft by 40 ft. Make a scale drawing of the courtyard using the following scales.

26. 1 cm : 1 ft 27. 1 cm : 5 ft 28. 1 cm : 10 ft

29. Given that �ABC ∼ �DEF,find the perimeter P and area A of �DEF.

A

B

C 8 D F

E

12

P = 32 A = 40

30. Given that �RSV ∼ �RTU, 31. Given: A (-3, 3) , B (1, 7) , C (5, 5) , find the coordinates of S and D (-1, 5) , E (1, 4) the scale factor. Prove: �ABC ∼ �ADE

y

x

R(0 ,0) S T(16 ,0)

V(0, -6) U(0, -8)

Lesson

7-4

Lesson

7-5

Lesson

7-6

15 ft

10 ft

20 ft

P = 48; A = 90

(12, 0) ; 4 _ 3

XW = 30; WZ = 15RS = 12; ST = 10

3.75 8.4

26–28. Check students’ work.

Since EJ _ JH

= FG _ GH

, −−

EF � −−

JG

by the Conv. of the � Proportionality Thm.

23. Since KL _ LM

= KP _ PN

,

−−

LP � −−

MN by the Conv. of the � Proportionality Thm.

AD = √

�

8 = 2 √

�

2 , AB = √

�

32 = 4 √

�

2 , AE = √

�

17 ,

and AC = 2 √

�

17 . AD _ AB

= 2 √

�

2 _ 4 √

�

2 = 1 _

2 and AE _

AC = √

�

17 _ 2 √

�

17

= 1 _ 2 . Thus AD _

AB = AE _

AC and the similarity ratio is 1 _

2 .

Since the corr. sides are proportional, and ∠A � ∠A

by the Reflex. Prop. of �, �ABC ∼ �ADE by SAS ∼.

EPS15



1. Travel A map is a scale model of a real city. The scale on the map is 1 in.:30 mi. Two cities are 165 mi apart. How far apart will the cities be on the map? (Lesson 7-1)

2. Design The logo for the Tri-City Corporation shows two triangles. On a coordinate plane, the triangles have coordinates A(2, 0), B(0, 4), C(–6, 2) and D(3, 0), E(0, 6), F(–9, 3). Determine whether the triangles are similar. Support your answer by describing a transformation. (Lesson 7-2)

3. Recreation The sails on the sailboat below have the given dimensions. Use similar triangles to prove �ABC ∼ �DEF.(Lesson 7-3)

18 6

5

15

D

F E

B C

A

4. Graphics A photograph shows a smaller version of the real item. The height of the Washington Monument is approximately555 ft. The monument in a photo is 5 in. tall. What is the scale factor of the actual monument to the monument in the photo? (Lesson 7-3)

5. Geography Riverside Park has campsites available for rent. Lot A has 50 ft of street frontage and 80 ft of river frontage. Find the river frontage for lots B, C, and D. (Lesson 7-4)

80 ft

50 ft 32 ft

40 ft 75 ft

A B C D

6. Architecture An amphitheater is being built according to the design shown. If the total footage on the right of the rows of seats is 232.5 ft, find the length of each section. (Lesson 7-4)

E D

C

B

A

15 ft25 ft

30 ft

40 ft

45 ft

7. Jake wants to know the height of the oak tree in his front yard. He measured his height as 68 inches and his shadow as 34 inches. At the same time, the tree has a shadow of 5.5 feet. How tall is the tree? (Lesson 7-5)

8. Recreation The kiddie pool and the lap pool at Centerville Park are similar rectangles. The lap pool measures 25 ft wide by 48 ft long. The kiddie pool is 8 ft long. How wide is the kiddie pool to the nearest tenth? (Lesson 7-5)

9. Melissa is enlarging her 4-by-6 photo by 150%. Find the coordinates of the enlarged photo. (Lesson 7-6)

4 2

6

4

y

x

0

2

5.5 in.

111 ft : 1 in.

11 ft

4.2 ft

B = 51.2 ft; C = 64 ft; D = 120 ft

(0, 0) , (0, 9) ,

(6, 0) , (6, 9)



2. The triangles are similar; ABC can be mapped to DEF by a dilation: (x, y) → (1.5x, 1.5y)

EPA8

Chapter 7



Extra Practice


Write a similarity statement comparing the three triangles in each diagram.

1.

B

C D

A 2. E H

G

F

3.

L

K

M

J

Find the geometric mean of each pair of numbers. If necessary, give the answers in simplest radical form.

4. 3 and 9 5. 4 and 7 6. 1 _ 2

and 5

Find x, y, and z.

7.

P

3

8 y

z

x

N

P

R

Q

8.

12 6 x

z

y T

S

U

V 9.9

15 y

z

x

C

D

B

A

Write each trigonometric ratio as a fraction and as a decimal rounded to the nearest hundredth.

10. sin A 11. cos A 12. tan A

B

A

25

24

7

C Use a special right triangle to write each trigonometric ratio as a fraction.

13. cos 30° 14. sin 45° 15. tan 60°

Use your calculator to find each trigonometric ratio. Round to the nearest hundredth.

16. sin 38° 17. cos 47° 18. tan 21°

Find each length. Round to the nearest hundredth.

19. DE 20. GH 21. KL

D

9.4

F

E

37º

H

G

J

16.4

22º J K

L

18.3 47º

Use your calculator to find each angle measure to the nearest degree.

22. tan -1 (3.5) 23. sin -1 ( 1 _ 5

) 24. cos -1 (0.05)

Find the unknown measures. Round lengths to the nearest hundredth and angle measures to the nearest degree.

25.

Q

R

S 3.8

2.5

26. T

U

V

9.6

27º 27.

X

W

Y 6.5

15

For each triangle, find the side lengths to the nearest hundredth and the angle measures to the nearest degree.

28. A (1, 4) , B (1, 1) , C (4, 1) 29. D (-3, 5) , E (-3, 1) , F (2, 5)

Lesson

8-1

Lesson

8-2

Lesson

8-3

3 √

�

3 2 √

�

7 √

�

10 _

2

7 _ 25

= 0.28 24 _ 25

= 0.96 7 _ 24

≈ 0.29

√

�

3 _

2

√

�

2 _ 2

√

�

3 _

1

0.62 0.68 0.38

15.6215.21 19.62

74° 12° 87°





EPS16

Chapter 8


Classify each angle as an angle of elevation or angle of depression.

30. ∠1 31. ∠2

4

1

3 2

32. ∠3 33. ∠4

Use a calculator to find each trigonometric ratio. Round to the nearest hundredth.

34. cos 127° 35. tan 131° 36. sin 114°

37. tan 158° 38. sin 85° 39. cos 161°

Find each measure. Round lengths to the nearest tenth and angle measure to the nearest degree.

40. AC 41. m∠E 42. m∠GA B

C

21

110º 37º

D

F

E

10

11

37º

43. m∠T 44. VX 45. BC

T

S U

7

15

9

V

W

X

12 16 38º

B C

A

3.8 4.2 25º

Write each vector in component form.

46. ⎯⎯⎯� AB with A (2, 3) and B (5, 6) 47. the vector with initial point C (3, 6) and terminal point D (2, 4)

48. ⎯⎯⎯� EF 49. ⎯⎯⎯⎯� GH

2 4

2

4

E

F

x

y

0

2 0

-4

x y

G

H

Draw each vector on a coordinate plane. Find its magnitude to the nearest tenth.

50. ⟨-3, 2⟩ 51. ⟨4, 3⟩ 52. ⟨2, -5⟩

Draw each vector on a coordinate plane. Find the direction of the vector to the nearest degree.

53. A wind velocity is given by the vector ⟨3, 4⟩.

54. The velocity of a rocket is given by the vector ⟨8, 1⟩.

Identify each of the following in the diagram.

55. equal vectors 56. parallel vectors 0

I

B A

H

E G

F C

-2

x

y

-2

Find each vector sum.

57. ⟨5, 0⟩ + ⟨-3, 6⟩ 58. ⟨-3, -1⟩ + ⟨0, -7⟩

59. ⟨1, 8⟩ + ⟨2, 3⟩ 60. ⟨-2, -1⟩ + ⟨-7, 9⟩

Lesson

8-4

Lesson

8-5

Lesson

8-6

30. angle of elevation32. angle of elevation

-0.60 -1.15 0.91

-0.40 1.00 -0.95

13.4 33°13°

139°9.9 1.8

⟨3, 3⟩ ⟨-1, -2⟩

⟨2, 4⟩ ⟨6, -3⟩

⎯⎯⎯

� CE �

⎯⎯⎯

� FG �

⎯⎯⎯

� HI

31. angle of depression33. angle of depression


⎯⎯⎯ � CE =

⎯⎯⎯

� FG

⟨2, 6⟩ ⟨-3, -8⟩

⟨3, 11⟩ ⟨-9, 8⟩

EPS17



1. Diving To estimate the height of a diving platform, a spectator stands so that his lines of sight to the top and bottom of the platform form a right angle as shown. The spectator’s eyes are 5 ft above the ground. He is standing 15 ft from the diving platform. How high is the platform? (Lesson 8-1)

15 ft

5 ft

2. Recreation A neighborhood park has a 15-foot-long space available to install a playground slide. If the maximum height of the slide is 6 ft, what are the lengths of the slide x and ladder y that should be installed? Round to the nearest tenth of a foot.(Lesson 8-1)

6 ft y x

15 ft

3. Building The escalator at the mall forms a 35° angle with the floor. The vertical distance from the bottom of the escalator to the top is 25 ft. How long is the escalator? Round to the nearest foot. (Lesson 8-2)

4. Sports A 3-foot-long skateboard ramp forms a 40° angle with the ground. How far above the ground is the end of the ramp? Round to the nearest foot. (Lesson 8-2)

5. Running A race includes a 0.25-mile hill on which runners travel from 510 ft of elevation to 570 ft of elevation. What angle does the hill form? Round to the nearest degree.(Lesson 8-3)

60 ft 0.25 mi

xº

6. Safety A lifeguard sees a swimmer struggling in the water at an angle of depression of 15°. The stand is 10 feet tall. What is the horizontal distance from the stand to the swimmer? Round to the nearest foot. (Lesson 8-4)

15º

10 ft

7. Aviation A helicopter pilot flying at an altitude of 1200 ft sees two landing pads directly in front of him. The angle of depression to the first landing pad is 40°. The angle of depression to the second pad is 28°. What is the distance between the two pads? Round to the nearest foot. (Lesson 8-4)

40º 28º

1200 ft

A B

8. Carpentry Sean is creating a triangular frame from three wooden dowels, which are 18 in., 12 in., and 15 in. long. What are the measures of each angle of the triangle? Round to the nearest degree. (Lesson 8-5)

9. Sports To estimate the width of the sand trap on a golf course, Matthew locates three points and measures the distances shown. What is the width, XZ, of the sand trap to the nearest foot? (Lesson 8-5)

140 yd

125 yd

47º

X

Z

Y

10. Recreation Jill swims due east across ariver at 2 mi/h. The river is flowing north at1.5 mi/h. What are Jill’s actual speed and direction? Round the speed to the nearest tenth and the direction to the nearest degree. (Lesson 8-6)

50 ft

13.4 ft; 6.7 ft

44 ft

2 ft

37 ft

827 ft

107 ft

2.5 mi/h; 37°, or N 53° E

41°, 56°, 83°

3°

EPA9

Chapter 8



Extra Practice


Copy each figure and the line of reflection. Draw the reflection of the figure across the line.

1. 2.

Reflect the figure with the given vertices across the given line.

3. A (-4, 1) , B (2, 4) , C (3, -2) ; x-axis 4. D (3, 1) , E (2, 4) , F (-2, 2) , G (2, -2) ; y = x

Copy each figure and the translation vector. Draw the translation of the figure along the given vector.

5.

� v

6.

⎯⎯⎯ � w

Translate the figure with the given vertices along the given vector.

7. A (-2, 1) , B (4, 3) , C (2, -2) ; ⟨2, 3⟩ 8. D (-1, 3) , E (2, 4) , F (3, 3) , G (3, -2) ; ⟨2, -2⟩

Copy each figure and the angle of rotation. Draw the rotation of the figure about the point P by m∠A.

9. 10.

P

A

A

P

Rotate the figure with the given vertices about the origin using the given angle of rotation.

11. A (2, 3) , B (-2, 1) , C (1, -1) ; 90° 12. D (-2, 3) , E (2, 4) , F (3, 1) , G (-2, 2) ; 180°

Draw the result of each composition of isometries.

13. Translate �ABC along � v and 14. Reflect �DEF across line m

then reflect it across line �. and then translate it along � w .

A B

C

� v

�

FE

Dm

⎯⎯⎯ � w

15. Copy the figure and draw two lines of reflection that produce an equivalent transformation.

IG

H

I�G�

H�

Lesson

9-1

Lesson

9-2

Lesson

9-3

Lesson

9-4


EPS18

Chapter 9



Describe the symmetry of each figure. Copy the shape and draw all lines of symmetry. If there is rotational symmetry, give the angle and order.

16. 17. 18.

Tell whether each figure has plane symmetry, symmetry about an axis, or neither.

19. 20. 21.

Copy the given figure and use it to create a tessellation.

22. 23. 24.

Classify each tessellation as regular, semiregular, or neither.

25. 26. 27.

Copy each figure and center of dilation P. Draw the image of the figure under a dilation with the given scale factor.

28. scale factor: 3 29. scale factor: 2 _ 3

P

P

Draw the image of the figure with the given vertices under a dilation with the given scale factor centered at the origin.

30. A (1, 3) , B (1, 5) , C (4, 3) ; scale factor 2

31. E (-2, 2) , F (2, 4) , G (4, -2) ; scale factor - 1 _ 2

Lesson

9-5

Lesson

9-6

Lesson

9-7

plane, axis plane, axis

plane, axis

regular

neithersemiregular





EPS19

Chapter 9



1. Transportation Two towns are located on the same side of a river. Two roads are being built to meet at the same point P on the river. Draw a diagram that shows where P should be located in order to make the total length of the roads as short as possible. (Lesson 9-1)

AB

2. Fashion A piece of fabric used for a scarf has a repeating pattern of trapezoids. To create the pattern, translate the trapezoid with vertices (-1, 3) , (3, 3) , (4, 1) , (-2, 1) along the vector ⟨0, -2⟩. Repeat to generate a pattern. What are the vertices of the third trapezoid in the pattern? (Lesson 9-2)

3. Computers A screen saver moves an icon around a screen. The icon starts at (20, 0) , and then it is rotated about the origin by 50°. Give the icon’s next position. Round each coordinate to the nearest tenth. (Lesson 9-3)

4. Recreation A hole at a miniature golf course has a barrier between the tee T and the hole H. Copy the figure and draw a diagram that shows how to make a hole in one. (Lesson 9-3)

T H

5. Sports A team’s Web site shows a baseball moving across the screen. The ball is reflected over line � and is then reflected over line m. Describe a single transformation that moves the ball from its starting point to its final position. (Lesson 9-4)

m

�

45˚

Agriculture Use the following information for Exercises 6–8.

Cattle ranchers brand their cattle to show ownership. Three different brands are shown. (Lesson 9-5)

Double A Bar O Bar Rocking R

6. Which brands have rotational symmetry?

7. Which brands have line symmetry?

8. Which capital letters could be used to create a brand with rotational symmetry?

Interior Design Use the following information for Exercises 9–11.

Three kitchen backsplash tile patterns are shown. Identify the symmetry in each pattern. (Lesson 9-6)

9.

10.

11.

12. Hobbies Reid has a baseball card that is 2.5-by-3.5 inches. He wants to enlarge it to poster size using a scale factor of 8. What size poster frame should he buy? (Lesson 9-7)

13. Hobbies A 40 in. by 30 in. piece of art is being made into a 1 in. by 3 __ 4 in. postage stamp. What scale factor should be used to reduce the art? (Lesson 9-7)

a 90° rotation

Bar O Bar

Double A, Bar O Bar

H, I, N, O, S, X, Z

translation

translation and glide reflection

translation and glide reflection

1 _ 40

2. (-1, -1) , (3, -1) , (4, -3) , (-2, -3) 3. (12.9, 15.3) 12. 20 in. by 28 in.



EPA10

Chapter 9



Extra Practice


Find each measurement.

1. the area of the parallelogram 2. the perimeter of the rectangle in which A = 15 x 2 ft 2

10 in.

6 in. 3x ft

3. b 2 of the trapezoid 4. the area of the kitein which A = 35 ft 2

5 ft

8 ft 10 m

8 m 17 m

5. the base of a triangle in which h = 9 and A = 135 in 2

6. the area of a rhombus in which d 1 = (3x + 5) cm and d 2 = (7x + 4) cm

Find each measurement.

7. the circumference 8. the area of �D of �C in terms of π

C

24 m in terms of π

D

5x ft

9. the circumference of �F in which A = 49 x 2 π cm 2

10. the radius of �E in which C = 36π in.

Find the area of each regular polygon. Round to the nearest tenth.

11. a regular hexagon with a side length of 8 in.

12. an equilateral triangle with an apothem of 5 √ � 3

_ 3

cm

Find the shaded area. Round to the nearest tenth, if necessary.

13.

8 ft

4 ft 14. 15.

8 m

24 m

8 yd

10 yd 4 yd

6 yd

Use a composite figure to estimate each shaded area. The grid has squares with side lengths of 1 in.

16. 17.

Lesson

10-1

Lesson

10-2

Lesson

10-3

11.5 x 2 + 23.5x + 10 cm 2

30 in.

168 m 2 6 ft

16x ft60 in 2

182.9 m 2

152 yd 2 82.3 ft 2

43.3 cm

166.3 in.

18 in.

14xπ cm

25 x 2 π ft 2 24π m

6 in 2 5 in 2

EPS20

Chapter 10


6 in. 8 in.

3 in.

10 in.

40 in.

25 in.

Estimate the area of each irregular shape.

18.

4

-4

4

-4

19.

4

-4

4

-4

Draw and classify the polygon with the given vertices. Find the perimeter and area of the polygon.

20. A (-2, 3) , B (0, 6) , C (6, 2) , D (4, -1) 21. E (-1, 3) , F (2, 3) , G (2, -1)

Find the area of each polygon with the given vertices.

22. R (-2, 3) , S (1, 5) , T (3, 1) , U (0, -2) 23. W (-4, 0) , X (4, 3) , Y (6, 1) , Z (2, -1)

Describe the effect of each change on the area of the given figure.

24. The height of the rectangle with height 10 ft and width 12 ft is multiplied by 1 _ 2

.

25. The base of the parallelogram with vertices A (-2, 3) , B (3, 3) , C (0, -1) , D (-5, -1) is doubled.

Describe the effect of each change on the perimeter or circumference and the area of the given figure.

26. The radius of �E is multiplied by 1 _ 4

.

E 12 ft 27. The base and height of a rectangle with base 6 in.

and height 5 in. are multiplied by 3.

28. A square has a side length of 7 ft. If the area is tripled, what happens to the side length?

29. A circle has a diameter of 20 m. If the area is doubled, what happens to the circumference?

A point is chosen randomly on −−

AD . Find the probability of each event.

30. The point is on −−

AC . 31. The point is on −−

AB or −−

CD .

A B C D

4 8 2

32. The point is not on −−

BC . 33. The point is on −−

BD .

Use the spinner to find the probability of each event.

34. the pointer landing on green

35. the pointer landing on blue or red

36. the pointer not landing on orange

108˚ 90˚

90˚ 36˚

36˚

37. the pointer not landing on red or yellow

Find the probability that a point chosen randomly inside the rectangle is in each shape. Round to the nearest hundredth.

38. the equilateral triangle

39. the parallelogram

40. the circle

41. the part of the rectangle that does not include the circle, triangle, or parallelogram

Lesson

10-4

Lesson

10-5

Lesson

10-6

44 units 2 31.5 units 2


18.5 units 2 19 units 2

The area is doubled.

The circumference is multiplied by √

�

2 .

The side length is multiplied by √

�

3 .

24. The area is multiplied by 1 __ 2 .

27. The perimeter is multiplied by 3. The area is multiplied by 9.

30. 6 _ 7

3 _ 7

3 _ 7 5 _

7

3_10 1_

2 36. 9 _ 10

13 _ 20

0.03

0.03


0.11

0.83

EPS21



1. Recreation Kathy is making a kite with diagonals of lengths 30 inches and 20 inches. How many square inches of fabric will she need? (Lesson 10-1)

Agriculture Use the following information for Exercises 2 and 3. An acre is 43,560 square feet. (Lesson 10-1)

2. If a one-acre piece of land is a rectangle with a base of 100 ft, what is its height?

3. If a one-acre piece of land is a square, what is the length of each side? Round to the nearest tenth.

4. The garden shown is a regular hexagon with a circular fountain at the center. What is the area of the garden? Round to the nearest square foot. (Lesson 10-2)

15 ft

30 ft

5. Food A bakery has cheesecake pans with three diameters: 18 cm, 22 cm, and 26 cm. Find the area of the bottom of each pan. Round to the nearest square centimeter. (Lesson 10-2)

6. Recreation A track for a toy car is a 2 ft by2 ft square with a semicircle at each end. What is the distance around the track? Round to the nearest foot. (Lesson 10-3)

7. Art Jonas is painting the shape shown on his ceiling. If a quart of paint covers 75 square feet, will one quart be enough to paint the entire shape? Explain. (Lesson 10-3)

4 ft

Transportation Use the following information for Exercises 8 and 9. The graph shows the speed of a car versus time. The base of each square on the graph represents 10 minutes, and the height represents 10 miles per hour. (Lesson 10-4)

Time (min) 10 20 30 40

Sp

eed

(m

i/h

)

10

20

30

40

8. What is the area of one square on the graph?

9. Estimate the shaded area of the graph.

10. Art Rasha is cutting a mat for a poster with an area of 480 in 2 . To find the dimensions of the mat, she multiplies the dimensions of the poster by 1.2. To find the dimensions of the opening, she multiplies the dimensions of the poster by 0.9. What is the area of the remaining part of the mat? (Lesson 10-5)

11. Food A restaurant sells two sizes of pizzas. The smaller pizza has a 12-inch diameter. If the area of the larger pizza is twice the area of the smaller pizza, what is the diameter of the larger pizza? Round to the nearest inch. (Lesson 10-5)

12. Transportation A commuter train stops at a station every 3 minutes and stays at the station for 20 seconds. If you arrive at the station at a random time, what is the probability that you will have to wait more than one minute for a train? Round to the nearest hundredth. (Lesson 10-6)

13. Sports A skydiver is delivering the game ball for a baseball game. Suppose he lands at a random point on the field. What is the probability that he will not land on the pitcher’s mound? Round to the nearest hundredth. (Lesson 10-6)

18 ft

90 ft

435.6 ft

208.7 ft

10 ft

No, the area of the shape is about 77.3 ft 2 .

8. 1 2 _ 3 mi

302.4 in 2

17 in.

0.56

0.97

300 in 2

1631 ft 2

5. 254 cm 2 ; 380 cm 2 ; 531 cm 2

≈ 26 2 _ 3 mi

EPA11

Chapter 10



Extra Practice


Classify each figure. Name the vertices, edges, and bases.

1. A B

F C

D E

M

G H

J

K L

2. S

T

3. V

W

X Y

Z

Describe the three-dimensional figure that can be made from the given net.

4. 5. 6.

Find the volume of each figure. Round to the nearest tenth.

7.

8 cm

14 cm

8.

8 ft

3 ft 9. 6 in.

12 in.

10. The dimensions of a prism with B = 14 cm 2 and h = 8 cm are doubled. Describe the effect on the volume.

11. The dimensions of a cylinder with r = 6 cm and h = 4 cm are multiplied by 2 _ 3

. Describe the effect on the volume.

Lesson

11-1

Lesson

11-2

triangular prism

rectangular pyramid

cylinder


The volume is multiplied by 8.

The volume is multiplied by 8 _ 27

.


EPS22

Chapter 11


Find the volume of each figure. Round to the nearest tenth.

12.

9 ft

B = 36 ft2

13.

14 m

4 m

4 m

14.

9 ft

18 ft

15. The dimensions of a cone with r = 8 cm and � = 17 cm are multiplied by 1 _ 2

. Describe the effect on the volume.

16. The dimensions of a pyramid with B = 128 m m 2 and h = 56 mm are tripled. Describe the effect on the volume.

Find the surface area and volume of each figure. Give your answers in terms of π.

17.

3 in.

18. 12 mm 19.

A = 36π ft2

20. The radius of a sphere with r = 24 cm is multiplied by 1 _ 3

. Describe the effect on the surface area and volume.

21. The radius of a sphere with r = 15 mm is multiplied by 4. Describe the effect on the surface area and volume.

Lesson

11-3

Lesson

11-4

V = 108 ft 3

V = 74.7 m 3

V = 1526.8 ft 3

The volume is multiplied by 1 _ 8

.

S = 9π in 2 ; V = 4.5π in 3

S = 144π ft 2 ; V = 288π ft 3

The volume is multiplied by 27.

S = 432π m m 2 ; V = 1152π m m 3


EPS23



1. Food Cookie dough is rolled in the shape of a cylinder. How can the dough be sliced to make circular cookies? (Lesson 11-1)

2. Recreation The tent shown is in the shape of a pentagonal prism. If a wall is used to divide the tent into two rooms, what shapes could the wall be? (Lesson 11-1)

Recreation Use the following information for Exercises 3 and 4.

A cylindrical pool has a 10 ft diameter. (Lesson 11-2)

3. How many gallons of water are needed to fill the pool to a depth of 4 feet? Round to the nearest gallon. (Hint: 1 gallon ≈ 0.134 cubic feet.)

4. If the pool is filled to a depth of 4 feet, how much will the water weigh? Round to the nearest pound. (Hint: 1 gallon weighs about 8.34 pounds.)

5. Hobbies The greenhouse shown is in the shape of a cube with a square pyramid on top. What is the volume of the greenhouse? (Lesson 11-3)

16 ft

16 ft

16 ft

12 ft

6. Food A snow-cone cup has a 3-inch diameter and is 4 inches tall. Another snow-cone cup has a 4-inch diameter and is 3 inches tall. Which cup will hold more? (Lesson 11-3)

7. Sports The circumference of a size 3 soccer ball is 24 in. The circumference of a size 5 soccer ball is 28 in. How many times as great is the volume of a size 5 ball as the volume of a size 3 ball? (Lesson 11-4)

Slice parallel to the bases.

2344 gal

19,553 lb

5120 ft 3

the 4 in. diameter cup

about 1.6 times as great

rectangles or pentagons

EPA12

Chapter 11



Extra Practice


Identify each line or segment that intersects each circle.

1.

C

A

B

D

F

E

2.

K

G H

F

J

Find the length of each radius. Identify the point of tangency and write the equation of the tangent line at this point.

3.

4

4

-4

-4

x

y

0

K

L

4.

4

4

-4

-4

x

y

0

NM

The segments in each figure are tangent to the circle. Find each length.

5. PQ 6. WZ

QS

P

R

3x - 5

2x + 7

YW

X

Z

7a + 5

5a + 12

Find each measure. Round to the nearest tenth, if necessary.

7. m � FB 8. PQ

AF C

B

E

D40˚

25˚

P

N

Q

7x + 5

10x - 4

9. �T � �W. Find m∠VWX. 10. BD

S U

T

W

VX

(3x + 78)˚

(7x - 6)˚

B

C

D

A

37

Find the area of each sector or segment. Round to the nearest tenth.

11.

6 m75˚

K L

J

12.

R S

Q

8 in.

60˚

Lesson

12-1

Lesson

12-2

Lesson

12-3

chords: −−

AB , −−

CE ; tangent: � ⎯ � FE ; radii:

−− CD ,

−− DE ;

secant: � ⎯ � AB ; diam.:

−− CE

chords: −−

GF , −−

FJ ; tangent: � ⎯ � GK ; radii:

−− HG ,

−− HF ,

−− HJ ;

secant: � ⎯ � GF ; diam.:

−− JF

radius of �K: 1.5; radius of �L: 3; pt. of tangency: (0, 3) ; eqn. of line: y = 3

radius of �M: 1; radius of �N: 2; pt. of tangency: (-2, 0) ; eqn. of line: x = -2

31 29.5

155° 26

141° 14.3

23.6 m 2 5.8 in 2

EPS24

Chapter 12


Find each arc length. Give your answers in terms of π and rounded to the nearest tenth.

13. A

B

7 ft

75˚ 14. C

D9 m

125˚

Find each measure or value. Round to the nearest tenth, if necessary.

15. m∠ABD 16. x

C

D

A

B

50˚

85˚

100˚(6x - 6)˚

LH K

J

17. x 18. angle measures of HJKL

(8x - 7)˚

(6x + 5)˚

H L

KJ

(4x - 6)˚

(8x + 13)˚

(2x + 25)˚

19. m � DF 20. m∠JMK 21. m∠RTQ

E

D F

100˚

70˚

H

K

J

M

L38˚

128˚S

N Q

R

T

P75˚

140˚

22. x 23. m∠AFE 24. m � GL

x˚210˚F

B

A

E

CD

75˚

120˚30˚

G

LM

K

H

J

110˚

25˚40˚

Find the value of the variable. Round to the nearest tenth, if necessary.

25.

B

A

x

D

C

E 95

7

26.

T U V

R

S

x

9

812

27.

NM

L

Kx

7

12

Write the equation of each circle.

28. �A with center A (2, -3) and radius 6 29. �B that passes through (3, 4) and has center B (-2, 1)

Graph each equation.

30. (x + 3) 2 + (y - 4) 2 = 1 31. x 2 + (y + 4) 2 = 16

Lesson

12-4

Lesson

12-5

Lesson

12-6

Lesson

12-7

35 _ 12

π;

9.2 m

25 _ 4

π; 19.6 m

42.5° 16

m∠H = 53.4°; m∠J = 129.2°; m∠K = 126.6°; m∠L = 50.8°

140° 83° 57.5°

30° 112.5° 90°

12.6 15.6 13.6

6

(x - 2) 2 + (y + 3) 2 = 36(x + 2) 2 + (y - 1) 2 = 34


EPS25


Car Colors

A

B

CD

EF

G

HBlue20%

Silver20%

White25%

Green10%

Other10%

Black8%

Yellow2%

Red 5%

Baseball Card Collection

A

B

CD

1970s

1980s

1990s


1. Measurement There is a water tower near Peter’s house in the shape of a cylinder. He wants to find the diameter of the tank. Peter stands 25 feet from the tower. The distance from Peter to a point of tangency on the tower is 80 feet. What is the diameter of the tank? (Lesson 12-1)

2. Travel Pikes Peak is 14,110 feet above sea level. What is the distance from the summit to the horizon, to the nearest mile? (Hint: Earth’s radius ≈ 4000 mi) (Lesson 12-1)

Hobbies Use the circle graph to find each measure for Exercises 3–6 to the nearest degree.

Eric collects baseball cards. He has 85 cards from the 1970s, 95 cards from the 1980s, and 125 cards from the 1990s. (Lesson 12-2)

3. � AB 4. � AC

5. ∠CDB 6. ∠ADC

Data Use the circle graph to find each measure for Exercises 7–10 to the nearest degree.

The circle graph shows the color of cars in a parking lot at the mall. (Lesson 12-2)

7. � HG

8. � CD

9. ∠AJH

10. ∠FJE

Hobbies Use the following information to find each area to the nearest tenth for Exercises 11 and 12.

A sprinkler system has three types of sprinkler heads: a quarter circle, a semicircle, and a full circle. The sprinkler will spray a distance of 15 feet from the sprinkler head. (Lesson 12-3)

11. What is the area of the sector that will be watered by the quarter circle sprinkler head?

12. What is the area of the sector that will be watered by the semicircle sprinkler head?

Art Use the diagram to find each value for Exercises 13–15.

The diagram represents an engraving on a stained glass window. (Lesson 12-4)

13. x

14. y

15. m � FE

16. Astronomy Two satellites are orbiting Earth. Satellite A is 10,000 km above Earth, and satellite B is 13,000 km above Earth. How many arc degrees of Earth does each satellite see? (Lesson 12-5)

44º

32º

A

B

17. Entertainment A group of friends ate most of a pepperoni pizza. All that was left was a piece of crust. What was the diameter of the original pizza? (Lesson 12-6)

2 in.

6 in.

18. Safety Three small towns have agreed to share a new fire station. To make sure each town has equal response time, the station should be the same distance from each town. The three towns are located on a coordinate plane at (0, 0) , (6, 0) , and (0, 8) . At which coordinates should the station be built? (Lesson 12-7)

(14x + 6)º

(8y + 6)º(5y + 24)º

G

CB

A

F

D

E

231 ft

146 mi

148° 100°

112° 100°

176.7 ft 2

353.4 ft 2

A: 136°; B: 148°

6.5 in.

(3, 4)

13. 614. 615. 108°

7. 36° 8. 7.2° 9. 72°10. 18°

EPA13

Chapter 12



Extra Practice


1. When text messaging on a telephone, pressing a 3 types D, E, F, or 3. Pressing a 7 types P, Q, R, S, or 7. How many messages are possible by pressing a 3, a 7, and then a 3?

2. At a company, each employee has an ID that consists of 2 digits followed by a letter. The letters Q and X are not used. How many employee IDs are possible?

3. If there are 8 finalists in a talent show, how many ways can a winner and a runner-up be chosen?

4. Jim’s soccer team has 18 members. How many ways can the coach choose a right forward, a center forward, and a left forward?

5. Erin’s health club offers 7 types of aerobics classes. She plans to attend 4 classes this week. How many ways can she choose 4 classes that are all different?

6. Francesca can take 4 of her 14 books on a trip. How many ways can she choose them?

Two number cubes are rolled. Find each probability.

7. Both cubes roll the same number. 8. The sum is greater than 8.

9. The sum is 8 or less. 10. Both cubes roll even numbers.

11. What is the probability that a random 2-digit number is a multiple of 7?

12. What is the probability that a randomly selected day in January is after the 20th?

13. A mother is making different lunches for each of her 3 children. If each child grabs a lunch bag at random, what is the probability that all 3 children will get the correct bag?

14. A teacher writes MATHEMATICS on a piece of paper and then cuts out each letter and puts them all in a bag. She will draw two letters at random. What is the probability that she will select an M and an A?

The shaded region is vertically centered in the flag. Find each probability.

14 in.

35 in.

2 in. 15. a random point inside the flag is in the shaded region

16. a random point inside the flag is above the shaded region

A marble is drawn from a bag and then Marble Drawing Experiment

Color Times Drawn

Pink 12

Green 10

Blue 16

Yellow 12

its color is recorded in the table.

17. Find the experimental probability of drawing a blue marble.

18. Find the experimental probability of drawing a pink or a yellow marble.

Lesson

13-1

Lesson

13-2

80

2400

56

4896

35

1001

1 _ 6

5 _ 18

13 _ 18

1 _ 4

13 _ 90

11 _ 31

1 _ 6

4 _ 55

1 _ 7

3 _ 7

0.32 or 32%

0.48 or 48%

EPS26

Chapter 13

CC13_G_MESE734262_EM_EPSc13.indd EPS26 5/17/11 11:15:58 PM


Find each probability.

19. rolling a number greater than or equal to 4 on a number cube twice in a row

20. drawing a face card from a deck, replacing it, and drawing a number card

21. Two number cubes are rolled—one blue and one yellow. Find the probability that the yellow cube is even, and the sum is 7. Explain why the events are dependent.

The table shows the results of a schoolwide survey on the Homecoming Dance Location Survey

Girls Boys

Gymnasium 67 58

Cafeteria 53 37

homecoming dance. Find each probability.

22. A student who prefers the cafeteria is a girl.

23. A surveyed student is male and prefers the gymnasium.

A bag contains 18 beads—5 blue, 6 yellow, and 7 red. Determine whether the events are independent or dependent. Find the indicated probability.

24. selecting a yellow and then a blue bead when they are chosen with replacement

25. selecting a yellow and then a blue bead when they are chosen without replacement

The table shows the side dish chosen with the lunch plate and the supper plate at a diner on one day.

Salad Fries Broccoli Total

Lunch 26 47 9 82

Supper 42 29 34 105

Total 68 76 43 187

26. Make a table of the joint and marginal relative frequencies. Round to the nearest hundredth where appropriate.

27. If you are given that a customer ordered a lunch plate, what is the probability that fries were chosen as the side dish?

28. If you are given that a customer ordered broccoli with the meal plate, what is the probability that it was the supper plate?

29. A table was chosen at random in the cafeteria, and there were 2 freshmen, 5 sophomores, 7 juniors, and 2 seniors eating there. A student is chosen at random from the table. What is the probability of choosing a freshman or a senior?

The numbers 1–20 are written on cards and placed in a bag. Find each probability.

30. choosing a number less than 10 or choosing a multiple of 5

31. choosing 20 or choosing an odd number

32. In an apartment building with 50 residents, 16 residents have cats, 28 residents are students, and 9 of the students have cats. What is the probability that a resident is a student or has a cat?

33. There are 8 couples in a dance competition, and each of the 3 judges must pick the couple they believe should win. Suppose the judges picked randomly. What is the probability that at least 2 judges picked the same couple?

Lesson

13-4

Lesson

13-5

Lesson

13-3

0.589, or 58.9%

0.270, or 27%

independent; 5 _ 54

1 _ 4

3 _ 5

11 _

20

7 _ 10

11 _ 32

dependent; 5 _ 51

1 _ 4

30 _ 169



about 0.57

about 0.79

EPS27

Chapter 13

CC13_G_MESE734262_EM_EPSc13.indd EPS27 5/17/11 11:16:35 PM


Music Use the following information for Exercises 1 and 2.

Serialism is a form of music in which the composer arranges each of the 12 tones in an octave to form a musical phrase. (Lesson 13-1)

1. How many ways can the 12 tones of an octave be arranged?

2. How many different musical phrases could a composer create by arranging only 5 of the 12 tones of an octave?

3. Drama A drama class is performing the Greek tragedy Antigone, by Sophocles. Of the 15 students in the class, 6 will make up the chorus. How many different ways can the chorus be selected? (Lesson 13-1)

4. Holidays Of December’s 31 days, the 25th and the 31st are holidays. What is the probability that a randomly chosen day in December is not a holiday? (Lesson 13-2)

5. Games If Sara’s dart lands in a red equilateral triangle, she wins a prize. Each triangle has a base of 2 in. If all locations on the 12 in. diameter target are equally likely, what is the probability that Sara wins a prize? (Lesson 13-2)

Literature Use the following information for Exercises 6 and 7.

The works of Chilean poet Pablo Neruda have been published in many languages. The school library has copies of two of his books in both English and Spanish. The table shows how many times each book has been checked out. (Lesson 13-3)

Books Checked Out

Canto General Extravagario

English 23 27

Spanish 17 14

6. What is the probability that Canto General was checked out in Spanish?

7. What is the probability that a student who checked out a Pablo Neruda book selected Extravagario in English?

Basketball Use the following information for Exercises 8–11.

The table shows the numbers of points scored by the top three scorers in last week’s basketball tournament that were made as 1-point shots (free throws), 2-point shots, and 3-point shots. (Lesson 13-4)

1-point 2-point 3-point

Tina 11 38 21

Stella 7 24 42

Misha 17 46 6

8. What is the total number of points scored by the three girls?

9. What is the joint relative frequency that represents points scored by Misha as 2-point shots?

10. What is the marginal relative frequency of the points that were made as 3-point shots?

11. Given that a point was scored by a free throw, what it the probability that Misha scored that point?

Immigration Use the following information for Exercises 12 and 13.

A group of 100 immigrants was studied over a one-year period. During the study, 63 of the immigrants found jobs, and 14 returned to their country of origin. Of the immigrants who found jobs, 6 of them returned to their countries before the end of the study. (Lesson 13-5)

12. What is the probability that an immigrant found a job or returned to his or her country of origin?

13. What is the probability that an immigrant did not find a job or returned to his country of origin?

12! = 479,001,600

95,040

3,603,600

≈ 93.55%

≈ 0.046

42.5%

33.33%

71 _ 100

= 71%

51 _ 100

= 51%

0.486

0.325

0.217

212

EPA14

Chapter 13

CC13_G_MESE734262_EM_EPAc13.indd EPA14 5/17/11 11:13:21 PMExtra PracticeEPCH13

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