Chapter GEOM 3e ExPrac 716-739 10/2 6/05 6:57 PM Page 71 6 1 Extra Practice…tarantamath.pbworks.com/w/file/fetch/49667090/GETEE… · · 2018-02-19717 Chapter 1 Extra Practice:

716

716 Chapter 1 Extra Practice: Skills, Word Problems, and Proof

Extra Practice: Skills, Word Problems, and ProofChapterChapter

11Lesson 1-1

Find the next two terms in each sequence.

1. 12, 17, 22, 27, 32,c 37, 42 2. 1, 1.1, 1.11, 1.111, 1.1111,c 1.11111, 1.111111

3. 5000, 1000, 200, 40,c 8, 4. 1, 12, 123, 1234,c 12,345, 123,456

5. 3, 0.3, 0.03, 0.003,c 6. 1, 4, 9, 16, 25,c 36, 49

Draw the next figure in each sequence.

7. 8.

Lesson 1-2

Name the space figure that can be formed by folding each net.

9. 10. 11. 12.

Make (a) an isometric drawing and (b) an orthographic drawing for eachfoundation drawing. 13–16. See margin.

13. 14. 15. 16.

17. You can cut four of the lettered squares from the figure at the right and fold the remaining net to make a box that is open at one end. Write the letters of the squares you could remove to do this. List all the possibilities.

Lessons 1-3 and 1-4

Write true or false.

18. A, D, F are coplanar. true 19. and are coplanar. false

20. A, B, E are coplanar. true 21. D, A, B, E are coplanar. true

22. 6 false 23. plane ABC 6 plane FDE true

24. and are skew lines. true 25. and are skew lines. false

26. 6 false 27. D, E, and B are collinear. false

28. How many sets of four collinear points are there in a 4-by-4 geoboard as pictured at the right? 10 sets

29. and do not intersect but intersects in one point. Make asketch that shows this. See margin.

ABDC)

CDAB

*CF)*

DE)

*EB)*

AD)*

DF)*

BC)*EF)*

FC)

C

A

D

B

F E

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Rig

ht

3

2

2

Front

Rig

ht34 1

1

FrontR

igh

t

2

3 3

1

Front

Rig

ht4 1

85

0.0003, 0.00003

7–8. See margin.

cuberectangular prism

cylindertriangular prism

A, C, G, I; D, G, F, I; A, B, G, H; A, D, C, F; B, C, H, I; C, F, D, G; A, B, H, I

page 716 Chapter 1 ExtraPractice

7.

8.

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14.

15.

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RightTop

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Front

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RightTop

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Front

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16. 29. Sample:B

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RightFront

Right

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Front

GEOM_3e_ExPrac_716-739 10/26/05 6:57 PM Page 716

717

Chapter 1 Extra Practice: Skills, Word Problems, and Proof 717

Skills,Word Problem

s,and Proof

Lessons 1-5 and 1-6

Use the figure at the right for Exercises 30–35.

30. If BC = 12 and CE = 15, then BE = j. 27

31. j is the angle bisector of j. , lBCA

32. Algebra BC = 3x + 2 and CD = 5x - 10. Solve for x. 6

33. Algebra If AC = 5x - 16 and CF = 2x - 4, then AF = j. 8

34. m&BCG = 60, m&GCA = j, and m&BCA = j. 60, 120

35. m&ACD = 60 and m&DCH = 20. Find m&HCA. 40

36. Algebra In the figure at the right, m&PQR = 4x + 47. Find m&PQS. 31

37. Algebra Points A, B, and C are collinear with B between A and C.AB = 4x - 1, BC = 2x + 1, and AC = 8x - 4. Find AB, BC,and AC. 7, 5, 12

Lesson 1-7

Make a diagram larger than the given one. Then do the construction.

38. Construct &A so that m&A = m&1 + m&2.

39. Construct the perpendicular bisector of .

40. Construct the angle bisector of &1.

41. Construct so that FG = AB + CD.

Lesson 1-8

(a) Find the distance between the points to the nearest tenth.(b) Find the coordinates of the midpoint of the segments with the given endpoints.

42. A(2, 1), B(3, 0) 1.4; ( , ) 43. R(5, 2), S(-2, 4) 7.3; ( , 3)44. Q(-7,-4), T(6, 10) 19.1; (– , 3) 45. C(-8,-1), D(-5,-11) 10.4; (– , –6)46. J(0,-5), N(3, 4) 9.5; ( , – ) 47. Y(-2, 8), Z(3,-5) 13.9; ( , )

48. A map of a city and suburbs shows an airport located at A(25, 11).

An ambulance is on a straight expressway headed from the airport to GrantHospital at G(1, 1). The ambulance gets a flat tire at the midpoint M of .As a result, the ambulance crew calls for helicopter assistance.a. What are the coordinates of point M? (13, 6)b. How far does the helicopter have to fly to get from M to G? Assume all

coordinates are in miles. 13 mi

Lesson 1-9

Find the perimeter (or circumference) and area of each figure.

49. 50. 51. 52.

4 in.

3 m1 cm

2 cm

2 cm3 cm

14 in.

7 in.

AG

32

12

12

32

132

12

32

12

52

FG

AB

BA

DC

1 2

CG

A

B C

G

D

H

E

Fx2x2

x2x2

x2x2

x2x2

38–41. See margin.

42 in., 98 in.2

10 cm, 5 cm2

3π m, π m294

(12 ± 2π) in., (16 ± 2π) in.2

P

QR

S

(7x�4)�

36�

Skills,Word Problem

s,and Proofpage 717 Chapter 1 Extra

Practice

38.

39.

40.

41.

GF

BA

21

A


718



22Lessons 2-1 and 2-2

For Exercises 1–3, identify the hypothesis and conclusion of each conditional.

1. If you can predict the future, then you can control the future.

2. If Dan is nearsighted, then Dan needs glasses.

3. If lines k and m are skew, then lines k and m are not perpendicular.

4. Write the converse of each statement in Exercises 1–3. See margin.

For each of the statements, write the conditional form and then the converse of theconditional. If the converse is true, combine the statements as a biconditional.

5. The number one is the smallest positive square.

6. Rectangles have four sides.

7. A square with area 100 m2 has sides that measure 10 m.

8. Two numbers that add up to be less than 12 have a product less than 37.

9. Three points on the same line are collinear.

Lesson 2-2

Is each statement a good definition? If not, find a counterexample.

10. A real number is an even number if its last digit is 0, 2, 4, 6, or 8.

11. A circle with center O and radius r is defined by the set of points in a plane adistance r from the point O. yes

12. A plane is defined by two lines.

13. Segments with the same length are congruent. yes

For Exercises 14 and 15, write the two statements that form each biconditional. Tellwhether each statement is true or false. 14-15. See margin.

14. Lines m and n are skew if and only if lines m and n do not intersect.

15. A person can be president of the United States if and only if the person is acitizen of the United States.

Lesson 2-3

Using the statements below, apply the Law of Detachment or the Law of Syllogismto draw a conclusion.

16. If Jorge can’t raise money, he can’t buy a new car. Jorge can’t raise money.

17. If Shauna is early for her meeting, she will gain a promotion. If Shauna wakesup early, she will be early for her meeting. Shauna wakes up early.

18. If Linda’s band wins the contest, they will win $500. If Linda practices, her bandwill win the contest. Linda practices. Linda’s band will win $500.

19. If Brendan learns the audition song, he will be selected for the chorus. IfBrendan stays after school to practice, he will learn the audition song. Brendanstays after school to practice. Brendan will be selected for the chorus.

No; two skew lines are a counterexample.

No; the real number must be an integer.

Jorge can’t buy a new car.

Shauna will gain a promotion.

5-9. See margin.

1-3. See margin.

14. If lines m and n areskew, then lines m and ndo not intersect; true.If lines m and n do notintersect, then lines mand n are skew; false.

15. If a person can bepresident of the UnitedStates, then the personis a citizen of the UnitedStates; true.If a person is a citizen ofthe United States, then

the person can bepresident of the UnitedStates; false.


1. Hyp: You can predictthe future.Concl: You can controlthe future.

2. Hyp: Dan is nearsighted.Concl: Dan needsglasses.

3. Hyp: Lines k and m areskew.Concl: Lines k and mare not perpendicular.

4. If you can control thefuture, then you canpredict the future.If Dan needs glasses,then Dan is nearsighted.If lines k and m are notperpendicular, thenlines k and m are skew.

5. If a number is one, thenit is the smallestpositive square. If anumber is the smallestpositive square, then itis one. A number is oneif and only if it is thesmallest positivesquare.

6. If a figure is a rectangle,then it has four sides. Ifa figure has four sides,then it is a rectangle.

7. If a square has an areaof 100 m2, then it hassides that measure 10m. If a square has sidesthat measure 10 m, thenit has an area of 100 m2. A square hasan area of 100 m2 if andonly if it has sides thatmeasure 10 m.

8. If two numbers add upto be less than 12, thentheir product is lessthan 37. If two numbershave a product that isless than 37, then theyadd up to be less than12.

9. If three points are on the same line, then theyare collinear. If threepoints are collinear, then they are on thesame line. Three pointsare collinear if and only if they are on thesame line.


719


For Exercises 20–23, apply the Law of Detachment, the Law of Syllogism, or bothto draw a conclusion. Tell which law(s) you used.

20. If you enjoy all foods, then you like cheese sandwiches. If you like cheesesandwiches, then you eat bread If you enjoy all foods, then you eat bread;

21. If you go to a monster movie, then you will have a nightmare. You go to amonster movie. You will have a nightmare; Law of Detachment.

22. If Catherine is exceeding the speed limit, then she will get a speeding ticket.Catherine is driving at 80 mi/h. If Catherine is driving at 80 mi/h, then she isexceeding the speed limit. Catherine will get a speeding ticket; both laws.

23. If Carlos has more than $250, then he can afford the video game he wants.If Carlos worked more than 20 hours last week, then he has more than $250.If Carlos works 15 hours this week, then he worked more than 20 hourslast week.

Lesson 2-4

24. Algebra You are given that 2c2 = 2bc + with c ≠ 0. Show that4b = 4c – a by filling in the blanks.

a. 2c2 = 2bc + a. Given

b. 4c2 = 4bc + ac b. 9 and 9 Mult. Prop. of ≠, Distr. Prop.

c. 4c = 4b + a c. 9 and Distributive Property Mult. Prop. of ≠

d. 9 4c – a ≠ 4b d. Subtraction Property

e. 4b = 4c - a e. 9 Symm. Prop. of ≠

25. Algebra Solve for x. Show your work. Justify each step.

Given: bisects &1. See margin.

Lesson 2-5

Algebra Find the value of x.

26. 27. 28.

29. Given:&1 and &2 are complementary.&3 and &4 are complementary.

Prove: &5 _&6 See margin.

30. Prove or disprove the following statement.

If &APB and &CPD are vertical angles,&APB and &APE arecomplementary, and &CPD and &CPF are complementary, then &APE and&CPF are vertical angles. See margin.

5 1 2 34 6

2x� (5x � 5)�2x�

4x�

(3x � 14)� (2x � 10)�

PF)

P

E

F

G

(6x�10)�

(4x�2)�

ac2

ac2

Skills,Word Problem

s,and Proof

24 15 25

Law of Syllogism.

If Carlos works 15 hours this week, then he can afford thevideo game that he wants; Law of Syllogism.

x2x2

x2x2

x2x2

Skills,Word Problem


Practice

25. The two anglesformed by thebisector have equalmeasures, so

4x ± 2 ≠ 6x – 10.

4x ± 12 ≠ 6xAdd. Prop. of ≠

12 ≠ 2xSubt. Prop. of ≠

6 ≠ xDiv. Prop. of ≠

29. l2 and l3 are vert.ls, so l2 O l3 bythe Vert. ls Thm. l1and l2 are compl.and l3 and l4 arecompl. (Given), so l1O l4 by the Cong.Complements Thm.From the diagram, l5and l1 are suppl. andl4 and l6 are suppl.Therefore l5 O l6by the Cong.Supplements Thm.

30. Counterexample:

A

BP

C

E F

D


720


33Lesson 3-1

Find ml1 and then ml2. State the theorems or postulates that justify youranswers. 1–4. See margin.

1. 2. 3. 4.

5. Complete the proof.

Given: / 6 m, a 6 b

Prove: &1 _&5

Statements Reasons1. / 6 m, a 6 b 1. Given2. &1 _&2 a. __?__ Vert. ' Thm3. &2 and &3 are supplementary. b. __?__ Same-Side Int. ' Thm4. &3 and&4 are supplementary. c. __?__ Same-Side Int. ' Thm5. &2 _&4 d. __?__ _ Supplements Thm6. &1 _&4 e. __?__ Trans. Prop. of _7. &4 _&5 f. __?__ Vert. ' Thm8. &1 _&5 g. __?__ Trans. Prop. of _

Lessons 3-2 and 3-3

Refer to the diagram at the right. Use the given information to determine which lines, if any, must be parallel. If any lines are parallel, use a theorem or postulate to tell why.

6. &9 > &14 none 7. &1 > &9

8. &2 is supplementary to &3. 9. &7 > &10 none

10. m&6 = 60, m&13 = 120 11. &4 > &13 none

12. &3 is supplementary to &10. 13. &10 > &15

14. Given: / 6 m, a 6 b, a # /

Prove: a # b See margin.

Lesson 3-4

Use a protractor and a centimeter ruler to measure the angles and the sides of eachtriangle. Classify each triangle by its angles and sides.

15. 16. 17. 18.

1a

b,

m 2 3 4

1 , m a

b2 3

45

12

64�

12

58�

2

1

125�

1 2

46�

10. r s, Vert. ' Thm. and Conv.of Same-Side Int. ' Thm.n

8. c d, Conv. of Same-Side Int. ' Thm.n

obtuse; scaleneright; scalene

acute; isoscelesobtuse; isosceles

none

See right.

See right.

21

65

43

87

109

1413

1211

1615

s

d

c

r

c d, Conv. of Alt. Int. ' Thm.n


Corr. ' Post.r s, Conv. of n


1. ml1 ≠ 134; Same-SideInt. ' Thm. ml2 ≠ 46; Alt. Int. lThm.

2. ml1 ≠ 125; Corr. 'Post. ml2 ≠ 55; Same-SideInt. ' Thm.

3. ml1 ≠ 58; Alt. Int. lThm. ml2 ≠ 122; Same-SideInt. ' Thm.

4. ml1 ≠ 64; Alt. Int. lThm. ml2 ≠ 116; Same-Side Int. ' Thm.

14. a # a and a b meansthat a # b since a line #to one of two || lines is# to the other (Thm 3-11). a # b and a mmeans that b # m forthe same reason.

n

n


721

Skills,Word Problem

s,and Proof

19. Use the figure at the right.What is the relationship between and ? Justify your answer. See margin.

Lessons 3-4 and 3-5

Algebra Find the value of each variable.

20. 21. 22. 23.

Algebra Find the missing angle measures.

24. 25.

Lessons 3-6

Write an equation in point-slope form of the line that contains the given points. 26–29. See margin.

26. A(4, 2), B(6,-3) 27. C(-1,-1), D(1, 1) 28. F(3,-5), G(-5, 3) 29. K(5, 0), L(-5, 2)

Write an equation in slope-intercept form of the line through the given points.

30. H(2, 7), J(-3, 1) 31. M(-2, 4), N(5,-8) 32. P(0, 2), Q(6, 8) 33. K(5, 0), L(-5, 2)

Lessons 3-6 and 3-7

Graph each pair of lines and state whether they are parallel, perpendicular, orneither. Explain.

34. y = 4x - 8 35. 13y - x = 7 36. y = x + 2 37. = -x +

y = 4x - 2 7 - = x y = x - 1 3x - = 0

Without graphing, tell whether the lines are parallel, perpendicular, or neither.Explain.

38. 2x + 3y = 5 39. y =-2x + 7 40. 5x - 3y = 0 41. y = 3y + 85x - 10y = 30 x - 2y = 8 y = x + 2 x = 3x + 8

42. On a city map,Washington Street is straight and passes through points at (7, 13) and (1, 5).Wellington Street is straight and passes through points at (3, 24) and (9, 32). Do Washington Street and Wellington Street intersect? How do you know?

Lesson 3-8

Use the segments for each construction.

43. Construct a square with side length 2a. 43–45. See margin.

44. Construct a quadrilateral with one pair of parallel sides each of length 2b.

45. Construct a rectangle with sides b and a.

a

b

53

153 y4

3y2

32

35 y24

3

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x� x�

125�

125�

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65100

55, 55122, 122, 122, 122, 116, 116

y 5 65x 1 23

5 y 5 2127 x 1 4

7y 5 x 1 2 y 5 21

5x 1 1

neither perpendicular parallel perpendicular

No; the slopes of the lines are equal and the y-intercepts are different

x ≠ 25; y ≠ 19x ≠ 110; y ≠ 102; z ≠ 82

#; m1? m2 ≠ –1#; m1? m2 ≠ –1n; same slope

neither; not same slope and m1? m2 u –1

A C

B

D E

F

a�

a�

b�

b�


x2x2

x2x2

Skills,Word Problem


Practice

19. n ; by the Ext.lThm, mlBCD ≠ a ±b and mlCDF ≠ a ±b. Thus, mlBCD ≠mlCDF by the Trans. Prop. of ≠. n bythe Converse of theAlt. Int. ' Thm.

26. y – 2 ≠ – (x – 4) or

y ± 3 ≠ – (x – 6)

27. y ± 1 ≠ (x ± 1) or y – 1 ≠ (x – 1)

28. y ± 5 ≠ –1(x – 3) or y – 3 ≠ –1(x ± 5)

29. y ≠ – (x – 5) or y – 2 ≠ – (x ± 5)

43.

44.

45.

b

a a

b b

b b

aa

a

a

a

a

15

15

52

52

DF4

BC4

DF4

BC4


722



44Lesson 4-1

kSAT O kGRE. Complete each congruence statement.

1. &S > 9 lG 2. >9 3. &E > 9 lT

4. > 9 5. #ERG > 9 6. > 9

7. #REG > 9 8. &R > 9 lA

State whether the figures are congruent. Justify each answer. 9–12. See margin.

9. #ABF;#EDC 10. #TUV;#UVW 11. $XYZV;$UTZV 12. #ABD;#EDB

Lessons 4-2 and 4-3

Where possible, explain how you would use SSS, SAS, ASA, or AAS to prove thetriangles congruent. If not possible, write not possible. 13–20. See margin.

13. 14. 15. 16.

17. Given: > , bisects . 18. Given: &1 > &2,&3 > &4, > ,

Prove: #PXZ > #PYZP is the midpoint of

Prove: #ADP > #BCP

19. Given: &1 > &2,&3 > &4, > 20. Given: 6 , 6 , >

Prove: #ABP > #DCP Prove:#MQP > #NRS

P

M RQ N

S

1 23 4

P

A DB C

1 23 4

NQMRPQRSNSMPDPAP

P

D

A

C

B

13

24

X

Z

Y

P M

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C O

AN

AP

L R

A

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P

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BA

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X V

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T

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R E

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kATS


9. Yes; corr. sides andcorr. ' are O.

10. No; the only known corr.O part is .



13. lT O lS, lY O lW,O ; ASA

14. O , O , lEAL O lPLA; SAS

15. not possible

16. O , O ,O ; SSS

17. bisects meansthat O . It isgiven that O ,and O by theRefl. Prop. of O. Thus,kPXM O kPYM by SSSand lXPM O lYPM byDef. of O. O bythe Refl. Prop. of O, sokPXZ O kPYZ by SAS.

18. It is given that ml1 ≠ml2 and ml3 ≠ ml4.By the l Add. Post.,mlDPA ≠ mlCPB.Since P is the midpt. of

, O . It is giventhat O , so kADPO kBCP by SAS.

19. l1 O l2 is given. lABPO lDCP by the OSuppls. Thm. l3 O l4and O are given.kABP O kDCP by AAS.

20. n and nmean that l1 O l4 andl2 O l3, respectively,by the Alt. Int. ' Thm. Itis given that MR ≠ NQ,so MR ± RQ ≠ NQ ±RQ, or MQ ≠ NR, by theAdd. Prop. of ≠.Therefore, kMQP OkNRS by ASA.

PQRSNSMP

DPAP

PCPDPBPAAB

PZPZ

PMPMPYPX

YMXMXYZP

CAACOCNAAOCN

LAALPLEA

SWTY

UV


723

Lesson 4-4

Explain how you would use SSS, SAS, ASA, or HL with CPCTC to prove eachstatement. 21–25. See back of book.

21. &MLN > &ONL 22. > 23. >

24. Given: &1 > &2,&3 > &4, 25. Given: PO =QO,&1 > &2,M is the midpoint of Prove:&A > &B

Prove:#PMQ > #RMQ

Lesson 4-5

Algebra Find the value of each variable.

26. 27. 28.

29. Given: &5 > &6, > 30. Given: 6 , >Prove: #PAB is isosceles. Prove:#QCD is isosceles.

Lessons 4-6 and 4-7

Name a pair of overlapping congruent triangles in each diagram. State whether thetriangles are congruent by SSS, SAS, ASA, AAS, or HL.

31. 32. 33.

34. Given: M is the midpoint of , 35. Given: #APQ > #BQP# , # ,&1 > &2 # , #

Prove: #ACM > #BDM Prove: X is the midpoint of .

1 265

43A B

P

X

Q

1 2

A M B

C D

AQPQBQPQAPBDMDACMC

AB

A

O

L

P N

MQ

M

W

S

RA

R O

F

G

P

Q

A

C D

B

P5 6

1 2 43A X Y B

PDPCBPAPPYPX

135�x� z�

y�57�

y�

x�

25�

x�

P

A

Q

O B1

2S

P

R

M Q

1

2

34

PR

M I

RB

T

O

S

E

M N

L ORIMBESTO

Skills,Word Problem

s,and Proof

65

x ≠ 57; y ≠ 66


x ≠ 45; y ≠ 90; z ≠ 45

kAON O kMOP; AASkRQM O kQRS; SSS

kARO O kRAF; HL


x2x2

34–35. See back of book.

Skills,Word Problem


Practice

29. O (Given)means that l1 O l2by the Isos. k Thm.l3 O l4 by the OSuppls. Thm. l5 Ol6 (Given), so kPXAO kPYB by ASA. O by CPCTC andkPAB is isos. by def.of isos.

30. O and O(both given).

lAPC O lBPD asvert. ', so kAPC OkBPD by SAS. lACPO lBDP becauseCPCTC. lPCD OlPDC by the Isos. kThm. mlACP ±mlPCD ≠ mlBDP ±mlPDC by the Add.Prop. of ≠, somlACD ≠ mlBDCby the l Add. Post.and substitution. O by the Conv. ofthe Isos. k Thm., andkQCD is isos. by def.of isos. k.

QCQD

PDPCBPAP

PBPA

PYPX


724


55Lesson 5-1


1. 2. 3. 4.

5. A sinkhole caused the sudden collapse of a large section of highway.Highway safety investigators paced out the triangle shown in the figureto help them estimate the distance across the sinkhole. What is thedistance across the sinkhole? 140 ft

Lessons 5-1 and 5-2

Algebra Use the figure at the right.

6. Find the value of x. 5

7. Find the length of . 7

8. Find the value of y.

9. Find the length of .

10. Given: > , > , 11. Given: bisects #BCN.> bisects #CBM.

Prove: A, B, and C are collinear. Prove:&A > &B

12. Find an equation in slope-intercept form for the perpendicular bisector of thesegment with endpoints H(-3, 2) and K(7,-5).

Lesson 5-3

Find the center of the circle that you can circumscribe about kABC.

13. A(2, 8) (1, 5) 14. A(-3, 6) (2, 2) 15. A(4, 3) (0, 0) 16. A(-10,-2) (–6, –6)B(0, 8) B(-3,-2) B(-4,-3) B(-2,-2)C(2, 2) C(7, 6) C(4,-3) C(-2,-10)

Is an angle bisector, altitude, median, or perpendicular bisector?

17. 18. 19. 20. B

A

B

A

B

AB

A

AB

y 5 107 x 2 61

14

A

M

B

C N

X

P

A

Q

C B

CQCPCX)

BQBPAQAP

152EG

52

AD

E FB

D

CA Gy + 5

2x - 3 x + 2

3y

5x

14

x3x

487x - 1

48

257

32 7 10

150 ft150 ft

182 ft

182 ft

280 ft

l bisector median

altitude


# bisector


x2x2

x2x2


10. It is given that each ofA, B, and C isequidistant from P andQ. By the Conv. of the #Bisector Thm, A, B, andC are on the same line,namely the # bisectorof .

11. Since X is on thebisector of lBCN andthe bisector of lCBM,X is equidistant from the sides , , and

(l Bisector Thm).Therefor X is equi-

distant from

(containing ) and

(containing ), thesides of lA. By theConv. of the l BisectorThm, X is on thebisector of lA.

CNS

ANS

BMSAMS

CNS

BCS

BMS

PQ


725

Skills,Word Problem

s,and Proof

21. Find the center of the circle that you can circumscribe about the triangle with vertices A(1, 3), B(5, 8), and C(6, 3). (3.5, 5.1)

22. Tell which line contains each point for #ABC.a. the circumcenter line kb. the orthocenter line mc. the centroid line /d. the incenter line n

23. Draw an acute triangle and construct its inscribed circle. See margin.

Lesson 5-4

Write (a) the inverse and (b) the contrapositive of each statement. 24–27. See margin.

24. If two angles are vertical, then they are congruent.

25. If figures are similar, then their side lengths are proportional.

26. If a car is blue, then it has no doors.

27. If a triangle is scalene, then it is not equiangular.

28. Suppose you know that &A is an obtuse angle in #ABC. You want to provethat &B is an acute angle. What assumption would you make to give an indirectproof? lB is not an acute angle.

Write the first step of an indirect proof of each statement.

29. #ABC is a right triangle. Assume kABC is not a right k.

30. Points J, K, and L are collinear. Assume points J, K, and L are not collinear.

31. Lines / and m are not parallel. Assume lines < and m are not n.

32. $XYZV is a square. Assume ~XYZV is not a square.

Lesson 5-5

List the sides of each triangle in order from shortest to longest.

33. 34. 35. 36.

Can a triangle have sides with the given lengths? Explain.

37. 2 in., 3 in., 5 in. No; 2 ± 3 w 5. 38. 9 cm, 11 cm, 15 cm 39. 8 ft, 9 ft, 18 ft

40. In #PQR, m&P � 55, m&Q = 82, and m&R � 43. List the sides of the trianglein order from shortest to longest.

41. In #MNS, MN � 7, NS � 5, and MS � 9. List the angles of the triangle in orderfrom smallest to largest. lM, lS, lN

42. Two sides of a triangle have side lengths 8 units and 17 units. Describe thelengths x that are possible for the third side. 9 R x R 25

C

TA

38�

36�M D

46�

QP

J

B53�

60�

N

R S82� 44�

k ,

m

n

A C

B

, , NSRSRN

, , PRQRPQ

, , PJPBJB, , MDQDMQ

, , CATACT

Yes; 9 ± 11 S 15. No; 8 ± 9 w 18.


Skills,Word Problem


Practice

23. Check students’work. The D shouldshow three acute '.The constructionshould includebisectors of at leasttwo of the '. The finaldiagram should showan incircle (tangent tothe three sides of theD).

24. a. If two ' are notvert., then they arenot O.

b. If two ' are not O,then they are notvert.

25. a. If figures are notsimilar, then theirside lengths are notprop.

b. If their side lengthsare not prop., thenfigures are notsimilar.

26. a. If a car is not blue,then it has doors.

b. If a car has doors,then it is not blue.

27. a. If a triangle is notscalene, then it isequiangular.

b. If a triangle isequiangular, then itis not scalene.


726



66Lesson 6-1

Graph the given points. Use slope and the Distance Formula to determine the mostprecise name for quadrilateral ABCD. 1–4. See margin.

1. A(3, 5), B(6, 5), C(2, 1), D(1, 3)

2. A(-1, 1), B(3,-1), C(-1,-3), D(-5,-1)

3. A(2, 1), B(5,-1), C(4,-4), D(1,-2)

4. A(-4, 5), B(-1, 3), C(-3, 0), D(-6, 2)

Lesson 6-2

Algebra Find the values of the variables in each parallelogram.

5. 6. 7. 8.

9. Given: PQRS and QDCAare parallelograms.

Prove: AP = BS

10. Given:$ABCDM is the midpoint of .

Prove: 6

Lesson 6-3

Based on the markings, decide whether each figure must be a parallelogram.

11. 12. 13. 14.

15. Describe how you can use what you know about parallelograms to construct apoint halfway between a given pair of parallel lines. See margin.

16. Given:$ABCD# ,#

Prove: BXDY is a parallelogram.

ACDYACBX

A D

X

Y

B C

ADPM

CD

A D

MP

B C

Q

P

R D

CB

S

A

y � 15

5x

3x � 1

y(2y � 5)�(2x � 5)�

(4x � 5)�(y � 10)�

y�

8x� 7x�

x ≠ 12; y ≠ 84 x ≠ 30; y ≠ 55 x ≠ 8; y ≠ 25 x ≠ 1; y ≠ 7

yes

See margin.

yesno yes

5x � 1

2y � 9

y � 2

3x � 1

See margin.

See margin.

x2x2


1.

trapezoid

2.

rhombus

3.

parallelogram

4.

square

9. Since PQRS is a ~, itsopp. sides are n, so n

and n . SinceQDCA is a ~, n .Thus, n becausetwo lines n to the sameline are n. PABS is a ~ bydef. of ~, and AP ≠ BSsince opp. sides of a ~are O.

10. The diagonals of a ~bisect each other, so P isthe midpt. of . P and Mare midpts. of two sidesof kACD so, by the kMidseg. Thm, O .

15. Sample answer: Marktwo O segments on eachof the two n lines. Thetwo segments areopposite sides of a ~.Construct (draw) thediagonals of the ~. Thediagonals intersect attheir midpts., which is the

ADPM

AC

ABPSQRAB

QRPSSBPA

�6 �4 �2 Ox

y

4

2

A

DC

B

3Ox

y

�3

1 A

D

C

B

4�2 Ox

yA

D

C

B�2

2 4 6Ox

yA

D

C

B6

4

2

desired point halfwaybetween the || lines.

16. ABCD is a ~ (given), son and O .

lBAX O lDCY by the Alt.Int. ' Thm. lAXB and

lCYD are rt. ' andtherefore O. kAXB OkCYD by AAS and O

by CPCTC. Since # and #(given), n . BXDY

has a pair of sides n andO, so BXDY is a ~.

DYBXACDYAC

BXDYBX

DCABDCAB


727


Lesson 6-4

For each parallelogram, determine the most precise name and find the measures ofthe numbered angles.

17. 18. 19. 20.

21. Use the information in the figure. Explain how 22. $ABCD is a rhombus. What is the relationship you know that ABCD is a rectangle. between &1 and &2? Explain.

Lesson 6-5

Find ml1 and ml2.

23. 24. 25. 26.

27. Suppose you manipulate the figure so that #PAB,#PBC,and #PCD are congruent isosceles triangles with their vertex angles at point P. What kind of figure is ABCD? Be sure to consider all the possibilities.

Lesson 6-6

Give coordinates for points D and S without using any new variables.

28. rectangle 29. parallelogram 30. rhombus 31. square

Lesson 6-7

32. For the figure in Exercise 31, use coordinate geometry to prove that themidpoints of the sides of a square determine a square. See margin.

33. In the figure,#PQR is an isosceles triangle. Points M and Nare the midpoints of and , respectively. Give a coordinate proof that the medians of isosceles triangle PQR intersect at H 0, . See margin.B2b

3A

PRPQ

D

S

x

y

(2a, 0)(�2a, 0)

D

S

x

y(0, b)

(c, 0)

D

S x

y(c � a, b)

(c, 0)O

D

S x

y(a, b)

A

B C

D

P

12

110�

1

2 70�1

267�

1

2110�

1

2A D

K

B CAP

D

B

C

12

3 4116�

1

2 3

4

55�

80�2 3

4

50� 1

1

2

Skills,Word Problem

s,and Proof

square; ml1 ≠ 45, ml2 ≠ 45


ABCD is an isos. trap. except when mlAPB ≠ 90,and in that case it is a square.

ml1 ≠ 110, ml2 ≠ 25 ml1 ≠ 90, ml2 ≠ 23 ml1 ≠ 110, ml2 ≠ 70ml1 ≠ 70, ml2 ≠ 70

D(0, b); S(a, 0) D(0, b); S(–a, 0) D(–c, 0); S(0, –b) D(0, 2a); S(0, –2a)

N

P(0, 2b)

Q(�2a, 0) R(2a, 0)

M

x

y

H(0, )2b3

Skills,Word Problem


Practice

18. rhombus; ml1 ≠ 50,ml2 ≠ 90, ml3 ≠ 40,ml4 ≠ 40

19. ~; ml1 ≠ 45, ml2 ≠45, ml3 ≠ 80, ml4≠ 55

20. rectangle; ml1 ≠116, ml2 ≠ 64, ml3≠ 32, ml4 ≠ 58

21. By the Conv. of theIsos. k Thm. and giventhat PA ≠ PB, it followsthat PD ≠ PA ≠ PB ≠PC. Thus, the diagonalsof ABCD bisect eachother (so ABCD is a ~)and are O (by the Seg.Add. Post.), so ABCD isa rectangle.

22. l1 and l2 are compl.The diags. ofrhombus ABCD are #and bisect each otherso the 4 small ks areO rt. ks. l1 andlCBK are compl.,and lCBK O l2, sol1 and l2 are compl.

32. Given: Square DRSQwith K, L, M, Nmidpts. of , ,

, and ,respectively. Prove:KLMN is a square.

K(a, a), L(a, –a),M(–a, –a), and N(–a,a) are midpts. of thesides of the squareKL ≠ LM ≠ MN ≠ NK≠ 2a. The slopes of

and areundefined. The slopesof and are 0,so adj. sides are # toeach other. Since all' are rt. ', the quad.is a rectangle. Arectangle with all Osides is a square.

NKLM

MNKL

D(0, 2a)

K(a, a)

R(2a, 0)

L(a, �a)

S(0, �2a)

M(�a, �a)

Q(�2a, 0)

N(�a, a)

y

xO

QDSQRSDR

33. The line through R(–a, 0)and M(–a, b) is

. The linethrough Q(–2a, 0) andN(a, b) is .

For each line, when x ≠0, , so the threemedians all contain point

.HA0, 2b3 B

y 5 2b3

(x 1 2a)y 5 b3a

(x 2 2a)y 5 b

23a


728



77Lesson 7-1

Algebra Solve each proportion.

1. = 10 2. = 36 3. = 2

4. = 6 5. = 6. = 21

Lesson 7-2

Algebra The polygons are similar. Find the values of the variables.

7. 8.

9. 10.

11. Are all equilateral quadrilaterals similar? Make a sketch to support your answer. No; sample sketch: a square and a rhombus

Lesson 7-3

Can you prove that the triangles are similar? If so, write a similarity statement and tell whether you would use AAM, SASM, or SSSM.

12. 13. 14.

15. 16. 17.

18. Refer to the figure at the right. Explain how you know that 6 .EDAB

12

5

46

10

8

CB

A

D

E

10

64.8 8

J

S

KT

RH

B

CA

D

E

F

RT

H15 24

1612

L

Z YE

X N W

M

Q T

21

148

12

C

P

3y

y3.57 x + 4

xy

85

2.5

30�

x�

5x y

4.5 37100�

3y - 2 y z 3x�

12 4

2

9x

37

92

x6

34

39

2x

612

x4

16x

49

x15

23

x ≠ ; y ≠ 6; z ≠ 163

803

x ≠ ; y ≠ 2Á103Á105

x ≠ 12; y ≠ 8x ≠ 30; y ≠ 4

Yes; kQCTM kMCP bySASM.

Yes; kABCM kEBD byAAM.

Yes; kXYZ M kPRQ by SSSM.

kCAB M kCED by SSSM. lA O lE ascorres. of M , so by theConv. of the Alt. Int. Thm.'

AB 6 ED>'

Yes; kHJK M kRST by SASM.

Yes; kXZYM kEWNby AAM.

no

x2x2

x2x2

7

6

4

9

10.5 6

P

R

QY

X Z


729


Lesson 7-4

Algebra Find the value of each variable. If an answer is not a whole number, leaveit in simplest radical form. 19–22. See margin.

19. 20. 21. 22.

23. Give a coordinate proof of the converse of Corollary 1 to Theorem 7-3. That is,prove that if is the altitude from C to side of #ABC, and if CD is thegeometric mean of AD and DB, then #ABC is a right triangle with its rightangle at C. See margin.

24. An artist is going to cut four similar right triangles from a rectangular piece ofpaper like the one shown below. What is the distance from B and D to thediagonal ?

Lesson 7-5


25. 26. 27. 28.

29. Suppose you are given a segment of length 1 unit and a segment of length x units.Show how you can apply the Side-Splitter Theorem to construct a segment of length .

30. The figure below shows the locations of a high school, a computer store, alibrary, and a convention center. The street along which the computer store andlibrary are located bisects the obtuse angle formed by two of the other streets.Use the information in the figure to find the distance from the library to theconvention center. 4.5 mi

1x

CD A B1

C Dx

AB

16

6

7x

40

2.1x21

x

10

13

9x

65

12 x

6013AC

ABCD

45

z y x

144 25

xzy

310

xy z

1 4

yx z

Skills,Word Problem

s,and Proof

725

11710

20

563

x2x2

See margin.

B

A

C

D12

5

3 mi

4 mi

6 miHighSchool

ComputerStore

ConventionCenter

Library

Skills,Word Problem


Practice

19. x ≠ ; y ≠ 2; z ≠2

20.

21. x ≠ 65; y ≠ 60; z ≠ 156

22. kABC M kEDC bySSSM, so lA O lE.Thus, O by theConv. of the Alt Int. 'Thm.

23. Place kABC in thecoordinate plane withA(–a, 0), B(b, 0), C(0,

) (given), and D(0, 0). Slope of ≠

. Slope of ≠

. The product ]

≠ ≠

–1, so # .

29.

Construct with AB ≠ xand BC ≠ 1. On anotherline from A, constuct oflength 1. Construct a linethrough C n to andintersecting in E. By theSide-Splitter Thm., , so DE ≠ .1

x

x1 5 1

DE

ADBD

AD

AC

A

B

x

1

1

C

E

D

BC4

AC4

ab2ab

!abb?!ab

2a

!abb

BC4!ab

a

AC4!ab

EDAB

z 5 "30y 5 "21;x 5 "70;

Á5Á5


730



88Lessons 8-1 and 8-2

Find the value of x. If your answer is not a whole number, leave it in simplestradical form.

1. 2. 3. 4.

5. A rectangular lot is 165 feet long and 90 feet wide. How many feet of fencing are needed to make a diagonal fence for the lot? Round to the nearest foot. 188 ft

Find the missing side lengths. Give answers in radical form if necessary.

6. 7. 8. 9.

Lessons 8-3 and 8-4

Find the value of x. Round lengths of segments to the nearest tenth and anglemeasures to the nearest degree.

10. 11. 12. 13.

14. 15. 16. 17.

18. An architect includes wheelchair ramps in her plans for the entrance to a new museum. She wants the angle that the ramp makes with level ground to measure 4°.Will the dimensions shown in the figure work? If not,what change should she make?

19. A 12-ft ladder is propped against a vertical wall. The top end is 11 ft above theground. What is the measure of the angle formed by the ladder with theground?

20. How long is the guy wire shown in the figure if it is attached to the top of a 50-ft antenna and makes a 70° angle with the ground? Round to the nearest tenth.

8 ft

100 ft

ramp

6.5

10x�

12

37�x

13

10

x�

7

6 x�

8

58�

32�x

8

54�

x11

23x�

548�

x

1

8 in.60�

30�26 cm 60�

30�

12 cm

6x

6

9 x60�

5

x9

12 x

5.6

13 cm, 13 cm!3

8 in., 16 in.!3

6 cm, 6 cm!2!2 units, units, 2 units!2!2

2911.0 9.4

4950 7.2

No; sample: In the plan, change 100 ft to 114.5 ft.not to scale

about 66.4°

53.2 ft

49

15 5!3 3!5 3!2

50 ftx

70�



27. a. –49, 142 , 38, 47

b. –11, 189

28. a. –118, –55 , 86,110

b. –32, 55

29. a. –54, 72 , –95,–33

b. –149, 39

30. a. –21, –56 , 27,–64

b. 6, –120lk

lklk

lk

lklk

lk

lklk

lk

lklk

731


Skills,Word Problem

s,and Proof

21. A 15-ft ladder is propped against a vertical wall and makes a 72° angle with theground. How far is the foot of the ladder from the base of the wall? Round tothe nearest tenth. 4.6 ft

Lesson 8-5

Solve each problem. Round your answers to the nearest foot.

22. A couple is taking a balloon ride. After 25 minutes aloft, they measure theangle of depression from the balloon to its launch place as 168. They are 180 ftabove ground. Find the distance from the balloon to its launch place. 653 ft

23. A surveyor is 300 ft from the base of an apartment building. The angle ofelevation to the top of the building is 248, and her angle-measuring device is 5 ftabove the ground. Find the height of the building. 139 ft

24. Oriana is flying a kite. She lets out 105 ft of string and anchors it to the ground.She determines that the angle of elevation of the kite is 488. Find the height thekite is from the ground. 78 ft

25. Two office buildings are 100 ft apart. From the edge of the shorter building, the angle of elevation to the top of the taller building is 28°, and the angle of depression to the bottom is 42°. How tall is each building? Round to the nearest foot. 90 ft; 143 ft

26. A plane flying at 10,000 ft spots a hot air balloon in the distance. The balloon is 9000 ft above ground.The angle of depression from the plane to the balloon is 308. Find the distance from the plane to the balloon. 2000 ft

Lesson 8-6

(a) Describe each vector as an ordered pair. Give the coordinates to the nearest unit. (b) Write the resultant of each pair of vectors as an ordered pair. 27–30. See margin.

27. 28. 29. 30.

Write the sum of the two vectors as an ordered pair.

31. �5, 9� and �-3, 2� 32. �-1, 0� and �4,-6� 33. �2, 4� and �0, 9� 34. �4,-2� and �-4, 2�

35. A helicopter lands 55 km west and 14 km north of the airport from which itdeparted. It followed a straight flight path. Find the magnitude and direction ofthe resultant vector �-55, 14�.

x

y

23�

70�

60 70x

y

53�

71�

100

90

x

y

38�140

130

25�

x

y

51�71�

150 60

100 ft

42�

28�

�2, 11� �3, -6� �2, 13� �0, 0�

about 56.8 km; about 14.3° north of west

W E

S

�55

14

N

Skills,Word Problem

s,and Proof


732



99Lesson 9-1

In Exercises 1–6, refer to the figure at the right.

1. What is the image of C 2. What rule describesunder (x, y) S (4,-2)? E the translation F S B? –2, 4

3. What is the image of H 4. What rule describesunder (x, y) S (-2, 4)? C the translation D S H? 4, –2

5. What is the image of C 6. What rule describesunder (x, y) S (-2,-4)? G the translation B S A? –8, 0

Use matrices to find the image of each figure under the given translation.

7. #ABC with vertices A(-3, 4), B(-1,-2), C(1, 5); translation: (x, y) S (-2, 5)

8. #EFG with vertices E(0, 3), F(6,-1), G(4, 2); translation: (x, y) S (1,-2)

9. #PQR with vertices P(-9,-4), Q(-5, 1), R(2, 8); translation: (x, y) S (-6,-7)

10. Write two translation rules of the form (x, y) S (x + a, y + b) that map the line y = x - 1 to the line y = x + 3.

Lesson 9-2

Given points S(6, 1), U(2, 5), and B(–1, 2), draw kSUB and its reflection image across each line.

11. y = 5 12. x = 7 13. y = -1 14. the x-axis

15. y = x 16. x = -1 17. y = 3 18. the y-axis

19. What are the two shortest words in the English language that you can write with capital letters so that each word looks like its own reflection across a line? A and I

20. The segments and are two different segments in the same plane. There is a translation such that is the translation image of . There is also a line kin the plane such that is the reflection image of across line k. If and are opposite sides of a quadrilateral, what kind of quadrilateral is it? rectangle

Lesson 9-3

Copy each figure and point P. Draw the image of each figure for the given rotationabout P. Label the vertices of the image. 21–24. See margin.

21. 608 22. 908 23. 458 24. 1808

25. The right triangle ABC shown here has side lengths 3, 4, and 5. Point P is the incenter of the triangle. Copy the triangle and draw the image of the triangle for a 60° counterclockwise rotation about P. See margin. B

P

C

A

C

B

P

D

H

S P

L

A

A

B

C

P

I H

P

GF

ArBrABABArBr

ABArBrArBrAB

lk

lk

lk 2

�2�2�4�6 2 4

x

y

F

HG

D

C

O

A B

E

4

A9(–5, 9), B9(–3, 3), C9(–1, 10)

E9(1, 0), F9(7, –4), G9(5, –1)

P9(–15, –11), Q9(–11, –6), R9(–4, 1)

Sample: (x, y) (x, y + 4), (x, y) (x - 4, y)SS

11–18. See back of book.


21.

22.

23.

24.

25.

PA�

B�

C �

A

BC

B�

� C �

D� � C

B

P

D

A�

L�

H �

S�

H

S P

L

A

AP

C

BA�

C �

B�

G� H �

F � I �

F G

I H

P

C



36.

37. Translate the polygonusing (x, y) U (x – 2,y – 5). Then dilatewith center (0, 0) andscale factor 3. Thentranslate using (x, y) U (x ± 2, y ± 5).

38.

40.

� m

Reflect over m, then over �.

� m

Reflect over �, then over m.

A�

B�

C �

D �

A D

BC

�2

�2

1 3 5

2

O x

y

733


26. What is the smallest angle of rotation you can use to have the rotation image ofthe figure below exactly overlap the original figure? 40°

Lesson 9-4

State what kind of symmetry each figure has.

27. 28. 29. 30.

31. Armando is going to draw a triangle that he will put on his backpack.a. If the triangle has a line of symmetry, what kind of triangle must it be? isoscelesb. If the triangle has two lines of symmetry, what kind of triangle must it be? equilateral

Lessons 9-5 and 9-6

The blue figure is the image of the gray figure. State whether the mapping is areflection, rotation, translation, glide reflection, or dilation.

32. 33. 34. 35.

36. The vertices of trapezoid ABCD are A(–1, –1), B(–1, 1), C(2, 2), and D(2, –1).Draw the trapezoid and its dilation image for a dilation with center (0, 0) andscale factor 3. See margin.

37. Suppose you know the coordinates of the vertices of a polygon. Describe howyou can use what you know about translations and dilations with respect to theorigin to find the coordinates of the vertices of the image polygon if the centerfor the dilation is (2, 5) and the scale factor is 3. See margin.

38. Find the image of the polygon for a reflection across line /followed by a reflection across line m. Then use a separate diagram to repeat the process, but reflect across line m first and then across line /. Each time, draw the intermediate image with dashed segments. See margin.

Lesson 9-7

39. Which of the four figures in Exercises 27–30 will tessellate a plane?

40. Use a square and an equilateral triangle to make a tessellation. The square andequilateral triangle should have congruent sides. See margin.

x

y

O

x

y

O

xy

OxyO

Skills,Word Problem

s,and Proof

rotation glide reflectiontranslation

all of them

dilation

line, rotation, point line, pointlineline

/ m

AB

C

D

EF

G

H

I

Skills,Word Problem

s,and Proof


734



1010Lesson 10-1

If possible, find the perimeter and area of each figure. If not possible, state why.

1. 2. 3.

4. 5. 6.

Lessons 10-2 and 10-3

Find the area of each trapezoid or regular polygon. Leave your answer in simplestradical form.

7. 8. 9. 10.

11. The patio section of a restaurant is a trapezoid 12. A mosaic design uses kite-shaped tiles with the with the dimensions shown in the figure. What dimensions shown in the figure. What is the is the area of the patio section? 3500 ft2 area of each tile? 2.625 in.2

13. The tiles for a bathroom floor are regular hexagons that are in. on each side.Find the area of an individual tile. Express the answer in radical form.

14. The floor of a gazebo is a regular hexagon with sides that are 9 ft long. What isthe area of the floor? Round to the nearest square foot. 210 ft2

Lesson 10-4

Find the ratio of the perimeters and the ratio of the areas of the blue figure to the red figure.

15. 16. 17.

18. A triangular banner has an area of 315 in.2. A similar banner has sides 1 times as long as those of the smaller banner. What is the area of the larger banner?

13

16 in.5 in.8 ft

6 ft

8 cm

5 cm

58

3 in.

1.75 in.

80 ft

50 ft

60 ft

4 ft

5 mm

4 in.

3 in.

6 in.

6 cm

18 in.

15 in.

26 yd

24 yd

13 m

17 m

12 m17 m

12 ft11 ft

13 ft6 m

3 m

2 m

5 m

5 ft

5 ft

4.33 ft

15 ft; 10.825 ft216 m; 12 m2

50 ft; 143 ft2

60 yd; 120 yd2 perimeter not possible as slantedsides could be any length; 270 in.247 m; 102 m2

72 cm215 in.2

mm2Á3254

32 ft2Á3

5 : 8; 25 : 64 3 : 4; 9 : 16 5 : 16; 25 : 256

73!3128

560 in.2

in.2



26. a. 6π cmb. 2π cm

27. a. 20π ft

b. π ft

28. a. 18π cm

b. π cm

29. a. 10π in.

b. π in.254

92

53

735


Skills,Word Problem

s,and Proof

19. You want to enlarge the picture on the front of a postcard by 10%. If theperimeter of the postcard is 44 cm, what will be the perimeter of the enlargement?

Lesson 10-5

Find the area of each polygon. Round your answers to the nearest tenth.

20. 21. 22. 23.

24. a regular hexagon with an apothem of 3 ft 31.2 ft2 25. a regular octagon with radius 5 ft 70.7 ft2

Lesson 10-6

(a) Find the circumference of each circle. (b) Find the length of the arc shown inred. Leave your answers in terms of π. 26–29. See margin.

26. 27. 28. 29.

30. A bicycle wheel has a radius of 0.33 m. How many revolutions does the wheel make when the bicycle is ridden 1 km? Round to the nearest whole number. 482

Lesson 10-7

Find the area of each shaded sector or segment. Leave your answers in terms of π.

31. 32. 33. 34.

35. A 14-in. diameter pizza is cut into 6 equal slices. About how many squareinches of pizza are in each slice? Round to the nearest square inch. 26 in.2

Lesson 10-8

Darts are thrown at random at each of the boards shown. If a dart hits the board,find the probability that it will land in the shaded area.

36. 37. 38. 39.

40. A square garden that is 80 ft on each side is surrounded by a cobblestone streetthat is 8 ft wide. If a child’s balloon lands at random in the region formed by thegarden and street, what is the probability that it lands on the street?

41. A dart hits the circular board shown in the figure at a random point. What isthe probability that it does not hit the shaded square? Express your answer interms of p.

75�

4 m

18 cm135�

6 in.

30� 30�

7 ft240�

5 in.

225�

9 cm150� 20 ft

120�

6 cm

8 m

10 m30�8 cm15 cm 46�

59�

14 in.

13 in.73�

7 ft9 ft

π ft2493

(12π – 9 ) in.2Á3π cm281

8

1136

(4π – 8) m2

14

13 1 – π4

724

30.1 ft2 20 m243.2 cm278.0 in.2

48.4 cm

5 cm18 c

m

1 – 2581p

Skills,Word Problem

s,and Proof


736



1111Lesson 11-1

The diagrams in Exercises 1–4 each show a cube after part of it has been cut away.Identify the shape of the cross section formed by the cut. Also, verify Euler’sFormula, F � V � E � 2, for the polyhedron that remains. 1–5. See margin.

1. 2. 3. 4.

5. The bases of the prism shown at the right are equilateral triangles.Make a sketch that shows how you can have a plane intersect the prism to give a cross section that is an isosceles trapezoid.


Find the (a) lateral area and (b) surface area of each figure. Leave your answers interms of π or in simplest radical form.

6. 7. 8. 9.

10. An optical instrument contains a triangular glass prism with the dimensions shown at the right. Find the lateral area and surface area of the prism. Round to the nearest tenth.

11. A company packages salt in a cylindrical box that has a diameter of 8 cm and a height of 13.5 cm. Find the lateral area and surface area of the box. Round to the nearest tenth.

Find the (a) lateral area and (b) surface area of each pyramid or cone. Assume that thebase of each pyramid is a regular polygon. Round your answers to the nearest tenth.

12. 13. 14. 15.


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

16. 17. 18. 19. 3 in.

5 in.

6 m

5 m

7 mm5 mm

5 mm

4 mm4 mm

3 mm

2 cm

6 cm

3 ft

5 ft

2 cm

6 cm

4 in.

7 in.

12 in.4 in.

10 in.

3 ft4 ft

6 ft

175 mm316 mm3

108 in.2; 144 in.2Á3Á340π in.2; 56π in.228π cm2; 36π cm284 ft2; 108 ft2

19.3 cm2; 21.3 cm2

339.3 cm2; 439.8 cm2

45π in.315π m3

56 in.2; 72 in.2 37.5 cm2; 47.9 cm2 55.0 ft2; 83.2 ft2 51.7 cm2; 71.0 cm2

2 cm

7 cm

2 cm

4 cm


1. equilateral k; 7 ± 10 ≠15 ± 2

2. rectangle; 7 ± 10 ≠15 ± 2

3. equilateral D; 7 ± 7 ≠12 ± 2

4. regular hexagon; 7 ± 10 ≠ 15 ± 2

5.


737


20. 21. 22. 23.

24. A greenhouse has the dimensions shown in the figure. What is the volume ofthe greenhouse? Round to the nearest cubic foot.

25. Find the volume of a can of chicken broth that has a diameter of 7.5 cm and aheight of 11 cm. Round to the nearest tenth. 486.0 cm3

26. A paper drinking cup is a cone that has a diameter of 2 in. and a height of

3 in. How many cubic inches of water does the cup hold when it is full to

the brim? Round to the nearest tenth. 5.7 in.3

Lesson 11-6

Find the volume and surface area of a sphere with the given radius or diameter. Give each answer in terms of π and rounded to the nearest whole number.

27. r = 5 cm 28. r = 3 ft 29. d = 8 in.

30. d = 2 ft 31. r = 0.5 in. 32. d = 9 m

The surface area of each sphere is given. Find the volume of each sphere interms of π.

33. 64p m2 m3 34. 16p in2 in.3 35. 49p ft2 ft3

36. A spherical beach ball has a diameter of 1.75 ft when it is full of air.What is the surface area of the beach ball, and how many cubic feet of air does it contain? Round to the nearest hundredth. 9.62 ft2; 2.81 ft3

Lesson 11-7

Copy and complete the table for three similar solids.

37.

38.

39.

40. How do the surface area and volume of a cylinder change if the radius and height are multiplied by ? S.A. is multiplied by . Volume is multiplied by .

41. For two similar solids, how are the ratios of their volumes and surface areas related? QV1

V2R2 5 QA1

A2R3

12564

1516

54

Ratio of Surface Areas Ratio of VolumesSimilarity Ratio

2 ; 3

■ ; ■

■ ; ■

■ ; ■

25 ; 64

■ ; ■

■ ; ■

■ ; ■

27 ; 64

343π6

32π3

256π3

12

12

7 in.

3 in.

6 ft

11 ft

10 m

10 m

13 m

12 cm10 cm

9 cm

Skills,Word Problem

s,and Proof

27. cm3, 524 cm3; 100π cm2, 314 cm2

28. 36π ft3, 113 ft3; 36π ft2, 113 ft2

29. in.3, 268 in.3; 64π in.2, 201 in.2

30. ft3, 4 ft3; 4π ft2, 13 ft2

31. in.3, 1 in.3; π in.2, 3 in.2

32. m3, 382 m3; 81π m2, 254 m2

243π2

π6

4π3

256π3

500π3

85

43

94

169

278

512125

540 cm3 863.9 in.3400 m3

339.3 ft3

5670 ft3

14 ft

8 ft 35 ft

10 ft

Skills,Word Problem

s,and Proof




1212Lesson 12-1

Algebra Assume that lines that appear to be tangent are tangent. P is the center ofeach circle. Find the value of x.

1. 2. 3. 4.

5. Given: Quadrilateral ABCD is circumscribed about �O.

Prove: AB +DC = BC +AD See margin.


Algebra Find the value of each variable. If your answer is not a whole number,round it to the nearest tenth. 10–12. See margin.

6. 7. 8. 9.

10. 11. 12. 13.

14. A polygon is inscribed in a circle. Are the perpendicular bisectors of the sidesof the polygon concurrent? Explain. See margin.

15. A circle has a diameter of 4 units. A chord parallel to a diameter is 1.5 units from the center of the circle. The endpoints of the diameter and the chord are the vertices of an isosceles trapezoid. What is the distance from the center of the circle to each leg of the trapezoid? Round to the nearest hundredth. 1.82 units

16. Given:&A and &D are inscribed angles in �O that intercept . and intersect at P.

Prove: #APB �#DPC

ACBDBC0

110�36�

b�c�

a�d�

80�

75�a�

b�32�

76�

a�

c� d�b�54�

77�

b�a�

20

9

9x

16

12x

x

12 3P

x

83

P xx

x64

10 P

x86

P x115�P x�

6510

6 2Á3

14.8

a ≠ 154; b ≠ 76

205.35.2

lA O lD since they both intercept .lBPA O lCPD because they are vertical .kAPB M kDPC by AA M.

'BC0

x2x2

x2x2

P

A

B

CD

S

R

QO

P

A

BC

DO

page 738 Chapter 12 Extra Practice

5. Tangents to a from apoint outside the areO, so AS ≠ AP, BP ≠BQ, CQ ≠ CR, and DR≠ DS. By the SegmentAdd. Post. and variousProps. or ≠,

AB ± DC ≠AP ± BP ± DR ± CR ≠AS ± BQ ± DS ± CQ ≠BQ ± CQ ± AS ± DS ≠BC ± AD

11. a ≠ 38; b ≠ 52; c ≠ 104;d ≠ 90

12. a ≠ 105; b ≠ 10013. a ≠ 55; b ≠ 72; c ≠ 178;

d ≠ 8914. Yes. Each side of the

polygon is a chord ofthe circle and the # bis.of any chord containsthe center of the circle.

((


page 739 Chapter 12 Extra Practice

17. x ≠ 193; y ≠ 60.518. 5.619. N 10.420. 7034. a circle of radius 5 cm,

concentric with theorig. circle

35. two rays n to and 2 cmfrom , and thesemicircle of radius 2cm with center A, opp.pt. B

36. a sphere of radius 1.5 in., and center Q

Q

1.5 in.

BA

2 cm

AB)

2 cm3 cm


Lesson 12-4

Algebra Assume that lines that appear to be tangent are tangent. Find the value ofeach variable. If your answer is not a whole number, round it to the nearest tenth.

17. 18. 19. 20.

21. 22. 23. 24.

25. The outer rim of a circular garden will be planted withthree colors of tulips. The landscaper has stretched two strings from a point P to help workers see how much of the circular rim should be planted with each color. Use the information in the figure at the right to find x and y.

26. Planks are placed across the circular pool shown inthe figure at the right. What is the length of the longest plank? 18 ft

Lesson 12-5

Write the standard equation for each circle with center P.

27. P = (0, 0); r = 4 x2 ± y2 ≠ 16 28. P = (0, 5); r = 3 x2 ± (y – 5)2 ≠ 9

29. P = (9,-3); r = 7 (x – 9)2 ± (y ± 3)2 ≠ 49 30. P = (-4, 0); through (2, 1) (x ± 4)2 ± y2 ≠ 37

31. P = (-6,-2); through (-8, 1) 32. P = (-1,-3); r = 3 (x ± 1)2 ± (y ± 3)2 ≠ 9

33. When a coordinate grid is imposed over a map, the location of a radio station isgiven by (113, 215). A town located at (149, 138) is at the outermost edge of thecircular region where clear reception is assured.a. Write an equation that describes the boundary of the clear reception region.b. If the radio station boosts power to increase the size of the clear-reception

region by a factor of 4, what will be the equation for the new boundary forclear reception?

Lesson 12-6

Draw and describe each locus. 34–36. See margin.

34. all points in a plane 3 cm from a circle with r = 2 cm

35. all points in a plane 2 cm from

36. all points in space 1.5 in. from a point Q

37. A dog is on a 20-ft leash. The leash is attached to a pipe at the midpoint of the back wall of a 30 ft-by-30 ft house, as shown in the diagram.Sketch and use shading to indicate the region in which the dog canplay while attached to the leash. Include measurements to describethe region. See margin.

AB)

8 ft

9 ft

12 ftx ft

120�

30�x�

y�

3 13 6

77

x

y

105�

20�x�165

6

x85�

140�

x�

y�

110�x�

6

12

x

10 8

7 x95�

72�

x�

y�

Skills,Word Problem

s,and Proofx N 5.6; y N 11.942.511.5x ≠ 112.5; y ≠ 67.5

(x ± 6)2 ± (y ± 2)2 ≠ 13

(x – 113)2 ± (y – 215)2 ≠ 852

(x – 113)2 ± (y – 215)2 ≠ 1702


90; 150

x2x2

30 ft

15 ft

15 ftdog

20 ft

Skills,Word Problem

s,and Proof

739


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