Geometry

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Write a similarity statement relating the three triangles in each diagram. See Problem 1.
PN
9. L 10. P Q 11. M
N O

JK SR

Algebra  Find the geometric mean of each pair of numbers. See Problem 2.

12. h4 samnd1110g mse_0704_t05271 13. 3hsamnd1418g mse_0704_t05272 14. 5hasnmd1112g5 mse_0704_t05273.ai
15. 7 and 9 16. 3 and 16 17. 4 and 49

Algebra  Solve for x and y. See Problems 3 and 4.
21. y
18. y x 19. 50 20. 4
40 x 9x
21
xy y 7
39

STEM 22. Architecture  The architect’s side-view drawing of Support post

ahThhosseaumlssteub1opi1sxpgt-ohsmrteytslspeeuo_hps0opt uio7ssr01te40ps_ohfttso0ttwa5plslho2.assH7impt4oioo.w1asnti1featgdhrm?afrtossmuep_tp0ho7er0ftrs4ot_nhtte0or5fot2ohh7fser5mid.a1gie1.gmse_0704B_edtr0o5omh2s7m6.1ai1gmse_0704_t05279

B Apply 23. a. The altitude to the hypotenuse of a right triangle divides

the hypotenuse into segments 2 cm and 8 cm long. Find 25 ft
Living room
the length of the altitude to the hypotenuse. Front Kitchen Back

b. Use a ruler to make an accurate drawing of the right

triangle in part (a).

c. Writing  Describe how you drew the triangle in part (b).

Algebra  Find the geometric mean of each pair of numbers.

24. 1 and 1000 25. 5 and 1.25 26. 18 and 12 27. 1 and 2 28. 128 and 17
2

29. Reasoning  A classmate says the following statement is true: The geometric mean
of positive numbers a and b is 1ab. Do you agree? Explain.

30. Think About a Plan  The altitude to the hypotenuse of a right triangle divides the

hypotenuse into segments with lengths in the ratio 1 : 2. The length of the altitude

is 8. How long is the hypotenuse?
• How can you use the given ratio to help you draw a sketch of the triangle?
• How can you use the given ratio to write expressions for the lengths of the

segments of the hypotenuse?
• Which corollary to Theorem 7-3 applies to this situation?

Lesson 7-4  Similarity in Right Triangles 465

31. Archaeology  To estimate the height of a stone figure, Anya 3.50 m
holds a small square up to her eyes and walks backward from
the figure. She stops when the bottom of the figure aligns with 1.84 m
the bottom edge of the square and the top of the figure aligns
with the top edge of the square. Her eye level is 1.84 m from the
ground. She is 3.50 m from the figure. What is the height of the
figure to the nearest hundredth of a meter?

32. Reasoning  Suppose the altitude to the hypotenuse of a right
triangle bisects the hypotenuse. How does the length of the
altitude compare with the lengths of the segments of the
hypotenuse? Explain.

The diagram shows the parts of a right triangle with h HSM11GMSE_0704_a02195
an altitude to the hypotenuse. For the two given a
measures, find the other four. Use simplest s1 s2 2nd pass 12-18-08
radical form. ᐍ1 Durk

ᐍ2

33. h = 2, s1 = 1 34. a = 6, s1 = 6 35. /1 = 2, s2 = 3 36. s1 = 3, /2 = 6 13

37. Coordinate Geometry  CD is the altitude to the hypotenuse of right △ABC.

The coordinates of A, D, and B are (4, 2), (4, 6), and (4h, 1s5m),1r1egspmesceti_v0el7y0. 4Fi_ntd10a5ll52

possible coordinates of point C.

Algebra Find the value of x. xϩ2 40. x 41. x 20
38. 39. x 18 12
xϩ5
xϩ3
12 x 5

Use the figure at the right for Exercises 42–43.

42. Prove Corollary 1 to Theorem 7-3. 43. Prove Corollary 2 to Theorem 7-3. C
Proo f Ghisvmen1: 1R gigmhst e△_A0B7C04w_itth0a5lh2ti8stum1d.a1ei1gmP rsooe f_G0i7v0en4:_  t R0ig5h2ht8s△m2A.a1Bi1Cgwmitshea_l0ti7tu0d4e_t052h8s3m.a1i1gmse_0704_t05284.ai

to the hypotenuse CD to the hypotenuse CD

Prove:  AD = CD Prove:  AB = AADC, AB = BC AD B
CD DB AC BC DB

44. Given:  Right △ABC with altitude CD to the hypotenuse AB y
Proo f Prove:  The product of the slopes of perpendicular lines is - 1. C

C Challenge 45. a. Consider the following conjecture: The product of the hsm11gmse_0704_t05286.ai
lengths of the two legs of a right triangle is equal to the
product of the lengths of the hypotenuse and the altitude b

to the hypotenuse. Draw a figure for the conjecture. Write A aD c B
O x
the Given information and what you are to Prove.
b. Reasoning  Is the conjecture true? Explain.

466 Chapter 7  Similarity

46. a. In the diagram, c = x + y. Use Corollary 2 to Theorem 7-3 to write two more ab
equations involving a, b, c, x, and y. xy

b. The equations in part (a) form a system of three equations in five variables. c

Reduce the system to one equation in three variables by eliminating x and y. hsm11gmse_0704_t0529
c. State in words what the one resulting equation tells you.

47. Given:  In right △ABC, BD # AC, and DE # BC. BE C
Proo f AD BE
Prove:  DC = EC

AD

Standardized Test Prep hsm11gmse_0704_t05293.ai

SAT/ACT 48. The altitude to the hypotenuse of a right triangle divides the hypotenuse into

segments of lengths 5 and 15. What is the length of the altitude?

3 5 13 10 5 15

49. A triangle has side lengths 3 in., 4 in., and 6 in. The longest side of a similar triangle
is 15 in. What is the length of the shortest side of the similar triangle?

1 in. 1.2 in. 7.5 in. 10 in.

Short 50. Two students disagree about the measures of angles in a kite. They know that two
Response angles measure 124 and 38. But they get different answers for the other two angles.
Can they both be correct? Explain.

Mixed Review

51. Write a similarity statement for the two triangles. R Q See Lesson 7-3.
How do you know they are similar? MN See Lesson 6-2.
P
Algebra  Find the values of x and y in ▱RSTV. V T
52. RP = 2x, PT = y + 2, VP = y, PS = x + 3 P

53. RV = 2x + 3, VT = 5x, TS = y + 5, SR = 4y - 1 RS

hsm11gmse_0704_t05295.ai

Get Ready!  To prepare for Lesson 7-5, do Exercises 54–56.

The two triangles in each diagram are similar. Find the value of x in each. See Lesson 7-2.

54. 30 cm 55. 56. 11 mm
15 cm 12 in.
84 cm 5 in. 7 inh. sm11gmse_067m0m4_t045m2m96.ai

x xx

hsm11gmse_0704_t05297.ai hsm11gmse_0704_t05298.ai hsm11gmse_0704_t05299.ai

Lesson 7-4  Similarity in Right Triangles 467

Concept Byte The Golden Ratio MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 7-4 Extends GM-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.isteimriailafroitry
ctriatenrgialefsotrotrsioalnvgelepsrotoblesomlvseapnrdotbolepmrosvaenrdeltaotipornosvheips in
activity rgeeloamtioentrsihcipfisguinregse. ometric figures.

MP 7

In his book Elements, Euclid defined the extreme and mean ratio using a proportion A CB

formed by dividing a line segment at a particular point, as shown at the right. In the hsm11gmse_0704fb_t05670.ai

diagram, C divides AB so that the length of AC is the geometric mean of the lengths of
AB ACCB . AC
AB and CB. That is, AC = The ratio CB is known today as the golden ratio, which is

about 1.618 ∶ 1.

Rectangles in which the ratio of the length to the width is the golden ratio are golden
rectangles. A golden rectangle can be divided into a square and a rectangle that is
similar to the original rectangle. A pattern of golden rectangles is shown at the right.

1

To derive the golden ratio, consider AB divided by C so that AB = AC .
AC CB

1. Use the diagram at the right to write a proportion that relates the A hsmx11gmsCe_07104fbB_t05671.ai
lengths of the segments. How can you rewrite the proportion as a hsm11gmse_0704fb_t05672.ai
quadratic equation?

2. Use the quadratic formula to solve the quadratic equation in Question 1.
Why does only one solution makes sense in this situation?

3. What is the value of x to the nearest ten-thousandth? Use a calculator.

Spiral growth patterns of sunflower seeds and the spacing of plant leaves on
the stem are two examples of the golden ratio and the Fibonacci sequence
in nature.

2

In the Fibonacci sequence, each term after the first two terms is the sum of the
preceding two terms. The first six terms of the Fibonacci sequence are 1, 1, 2, 3, 5,
and 8.

4. What are the next nine terms of the Fibonacci sequence?

5. Starting with the second term, the ratios of each term to the previous term for the

first six terms are 1 = 1, 2 = 2, 3 = 1.5, 5 = 1.666 . . ., and 8 = 1.6. What are the
1 1 2 3 5
next nine ratios rounded to the nearest thousandth?

6. Compare the ratios you found in Question 5. What do you notice? How is the
Fibonacci sequence related to the golden ratio?

468 Concept Byte  The Golden Ratio

Exercises

7. The golden rectangle is considered to be pleasing to the human eye. Of the
following rectangles, which do you prefer? Is it a golden rectangle?



Rectangle 1 Rectangle 4

Rectangle 2 Rectangle 3

8. A drone is a male honeybee. Drones have only one parent, a queen. Workers and

Pqhausremteon1fst1hagreemfafesmmei_lay0let7rhe0oe4hnsbsehm_yobtw10ei1e5nsg6g. m7Fthe3semeaa_nl0ecse7sh0htao4svrbmes_ot1wtf01ao5gdp6mra7orsne4ene_tiss0, s7ah0doh4rwosbnnm_ebt1a0e1nl5ogd6wma7, 5wqsuehe_eer0ne7.D04b_t05676

represents a drone and Q represents a queen.

Q DQQ D Great-Great-Grandparents

QD Q Great-Grandparents
Q D Grandparents

Q Parent

D Child

a. Continue the family tree for three more generations of ancestors.
b. Count the number of honeybees in each generation. What pattern

hsdmo 1yo1ug nmostiec_e?0704b_t05779

9. What is the relationship between the flowers and the Fibonacci sequence?



10. In △ABC, point D divides the hypotenuse into the golden ratio. That is, AD : DB is C
about 1.618 ∶ 1. CD is an altitude. Using the value 1.618 for AD and the value 1 for x

DB, solve for x. What do you notice? DB

A

hsm11gmse_0704b_t05677

Concept Byte  The Golden Ratio 469

Concept Byte Exploring Proportions CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
in Triangles
Use With Lesson 7-5 GM-ACFOS.D.9.12. GM-CaOke.4fo.1rm2 aMl gaekoemfoertmricalcognesotmruecttriiocns
wcointhstaruvcatiroientsy wofitthooalsvaarniedtymoefthtooodls .a.n.d
technology MmePth5ods . . .
MP 5

1

Use geometry software to draw △ABC. Construct point D on AB. Next, construct B
a line through D parallel to AC. Then construct the intersection E of the parallel
line with BC. D
E
1. Measure BD, DA, BE, and EC. Calculate the ratios BD and BE .
DA EC A

2. Manipulate △ABC and observe BD and BE . What do you notice?
DA EC

3. Make a conjecture about the four segments formed by a line parallel to one

side of a triangle intersecting the other two sides. C

2

Use geometry software to construct △ADE with vertices A(3, 3), D( -2, 0), and hsm11gmse_0705a_t05619
E(5, 1).
A
4. Measure AD, AE, and DE. Give your answers to the nearest tenth. E

5. Suppose B is the point on AD such that AB = 2 AD, and C is the point on AE D
3
such that BC } DE. Describe how you could approximate the coordinates of
point B.

6. Now use the geometry software to draw BC and manipulate the segment so
that it satisfies the conditions given in Exercise 5. What are the coordinates of
points B and C?

Exercises geom12_se_ccs_c07l05_t01.ai
7. Construct <AB> } <CD> } <EF>. Then construct two transversals that intersect all
three parallel lines. Measure AC, CE, BD, and DF . Calculate athnedrBDaDtFi.oMs ACaCEke AB
BD AC
and DF . Manipulate the locations of A and B and observe CE a CD
EF
conjecture about the segments of the transversals formed by the three parallel

lines intersecting two transversals.

8. Suppose four or more parallel lines intersect two transversals. Make a
conjecture about the segments of the transversals.

470 Concept Byte  Exploring Proportions in Triangles hsm11gmse_0705a_t05621

7-5 Proportions in Triangles CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

MG-ASFRST.9B1.42 .GPr-oSvReTt.h2e.4o rePmrosvaebtohuetotrreimansgalebso.u.t .a
tlirniaenpgalerasl.le.l.atolionneepsairdaelloelf taotorinaengsliedediovfidaetsritahnegle
doitvhiedretswthoeporothpeorrttiwonoapllryo.p.o.rtionally . . .

MP 1, MP 3, MP 4

Objective To use the Side-Splitter Theorem and the Triangle-Angle-Bisector Theorem

An artist uses perspective to draw parallel lampposts along a city
street, as shown in the diagram. What is the value of x? Justify
your answer.

Use what you 0.5 in. 0.42 in. 1.25 in.
know about similar
triangles to plan 1.42 in.
a pathway to
a solution. x
0.57 in.
MATHEMATICAL

PRACTICES

0705_a02197

The Solve It involves parallel lines cut by two transversals that intersect. In this lesson,
you will learn how to use proportions to find lengths of segments formed by parallel
lines that intersect two or more transversals.

Essential Understanding  When two or more parallel lines intersect other lines,
proportional segments are formed.

Theorem 7-4  Side-Splitter Theorem

Theorem If . . . Then . . .
If a line is parallel to one side <RS> } <XY>
of a triangle and intersects the XR = YS
other two sides, then it divides Q RQ SQ
those sides proportionally. R
S

X Y

hsm11gmse_0705_t05328 471

Lesson 7-5  Proportions in Triangles

Proof Proof of Theorem 7-4: Side-Splitter Theorem Q
Given:  △QXY with <RS> } <XY>
R3 4 S
Prove:  XR = YS
RQ SQ 1 2
X Y

Statements Reasons
1) <RS> } <XY> 1) Given

2) ∠1 ≅ ∠3, ∠2 ≅ ∠4 2) If lines are }, then corresphosnmdi1n1gg⦞masree_≅0.705_t05329
3) AA ∼ Postulate
3) △QXY ∼ △QRS
XQ YQ
4) RQ = SQ 4) Corresponding sides of ∼ △s are proportional.

5) XQ = XR + RQ, 5) Segment Addition Postulate

YQ = YS + SQ
XR + RQ YS + SQ
6) RQ = SQ 6) Substitution Property
7) Property of Proportions (3)
7) XR = YS
RQ SQ

Problem 1 Using the Side-Splitter Theorem

How can you use What is the value of x in the diagram at the right? M
the parallel lines in
the diagram? PK = NL Side-Splitter Theorem 12 9 3
KM LM
KL is parallel to one +1 K L 00000 0
side of △MNP. Use the x 12 = 9x Substitute. 11111 1
Side-Splitter Theorem to xϩ1 x 22222 2
set up a proportion. 9x + 9 = 12x Cross Products Property 33333 3
P N 44444 4
9 = 3x Subtract 9x from each side. 55555 5
66666 6
3 = x Divide each side by 3. 77777 7
88888 8
Grid in the number 3. 99999 9

Got It? 1. a. What is the value of a in the diagrahmsmat11gmse_0705a_t053132 0
the right?
b. Reasoning In △XYZ, RS joins XY and YZ aϩ4 hsm1118gmse_0705_t05394

with R on XY and S on YZ, and RS } XZ. If
YR = YS = 1, what must be true about RS?
RX SZ
Justify your reasoning.

hsm11gmse_0705_t05331

472 Chapter 7  Similarity

Corollary  Corollary to the Side-Splitter Theorem

Corollary If . . . Then . . .
If three parallel lines intersect a } b } c  a
two transversals, then the A W AB = WX
segments intercepted on the b B X BC XY
transversals are proportional.
cC Y

You will prove the Corollary to Theorem 7-4 in Exercise 46.

Problem 2 Finding a Length hsm11gmse_0705_t05395

What information Camping  Three campsites are shown in the 8 yd 6.4 yd
does the diagram diagram. What is the length of Site A along Site C
give you? the river?
The lines separating the
campsites are parallel. Let x be the length of Site A along the river.
Think of the river and
the edge of the road x = 79.2 Corollary to the Side-Splitter Theorem
as transversals. Then 8
the boundaries along Site A Site B
the road and river for 7.2x = 72 Cross Products Property 9 yd 7.2 yd
each campsite are
proportional. x = 10 Divide each side by 7.2.

The length of Site A along the river is 10 yd.

Got It? 2. What is the length of Site C along the road?

Essential Understanding  The bisector of an angle of a triangle divides the
opposite side into two segments with lengths proportional to the sides of the triangle
that form the angle.

Theorem 7-5  Triangle-Angle-Bisector Theorem

Theorem IAfD. >.b.isects ∠CAB Then . . .
If a ray bisects an angle of a A
triangle, then it divides the CD = CA
opposite side into two segments B DB BA
that are proportional to the CD
other two sides of the triangle.

You will prove the Triangle-Angle-Bisector Theorem in Exercise 47.

hsm11gmse_0705_t05396 473

Lesson 7-5  Proportions in Triangles

Problem 3 Using the Triangle-Angle-Bisector Theorem

Algebra  What is the value of x in the diagram at the right? R 10 Q 18 S

12 x
P
PQ bisects ∠RPS. Use the Triangle-Angle- RQ = PR
Bisector Theorem to write a proportion. QS PS

Substitute corresponding side lengths in the 10 = 12 hsm11gmse_0705_t05397
proportion. 18 x

Use the Cross Products Property. 10x = 216

Divide each side by 10. x = 21.6

Got It? 3. What is the value of y in the diagram at the right?

y 24
9.6 16

Lesson Check hsm11gmse_0705_t05398

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Use the figure to complete each proportion.
PRACTICES

6. Compare and Contrast  How is the Corollary to the

1. ab = ■ 2. b = e ab c Side-Splitter Theorem related to Theorem 6-7: If three
e ■ f de f
(or more) parallel lines cut off congruent segments on

3. b a = ■ one transversal, then they cut off congruent segments
+ +
c e f on every transversal?

What is the value of x in each figure? 7. Compare and Contrast  How are the Triangle-Angle-
Bisector Theorem and Corollary 1 to Theorem 7-3
4. 10 5. 12 16 alike? How are they different?

30 x hsm11gmse_0705_t05399 8. Error Analysis  A classmate says you can use the
45 x 20 Side-Splitter Theorem to find both x and y in the
diagram. Explain what is wrong with your classmate’s
statement.

hsm11gmse_0705_t05400hsm11gmse_0705_t05401 2y
3 x 2.4

7

474 Chapter 7  Similarity

Practice and Problem-Solving Exercises

A Practice Algebra  Solve for x. See Problem 1.

9. 8 6 10. x2 11. x 12.
3 8 12
10 x 13 Ϫ x xϩ5 9 2x
12 xϩ4

Marine Biology  Use the information shown on the auger shell. See Problem 2.

1134.. WWhshhmaatt1iiss1ttghhmee vvsaaell_uu0ee7oo0ff yx5??_t05h4s0m3 11gmse_0705_t054h0sm5 11gmse_0705_t05h4s0m611gmse_0705_t05404

Algebra Solve for x. 16. 6 y 7.5
15. 4 5 8.8 mm x
x 10 mm
45 11 mm

6x

17. 18. 8 12
2
hsm9 11gm1s1e_0705_t054072 hsm11gm2s4e_0705_t05408

x

4x

Algebra  Solve for x. See Problem 3.

19. hsm11xgmse_0705_2t00.5 409x hsm311gmse2_10.7 05_t058410 20 22. 4
6
12 14
58
5 x x
10 6

23. Writing  The size of an oil spill on the open ocean is difficult to measure

hsmlwdei1onr1eugcgtldhtlmyyo.o fsUueths_uee0stoe7hi?el0s5fpi_gilutl0rien5da4ti1rhteh1scemtlryi1.gW1htghtmaotdsmeese_ca0rsi7ub0ree5hm_oetwn0t5yso4au1nhc2dsomcual1dlc1fuignlamdtitoshnees_0705_t0A5h4s1m311gmBse_D0705_t05414

24. The lengths of the sides of a triangle are 5 cm, 12 cm, and 13 cm. Find C
the lengths, to the nearest tenth, of the segments into which the bisector E
of each angle divides the opposite side.

0705_a02215

Lesson 7-5  Proportions in Triangles 475

B Apply Use the figure at the right to complete each proportion. Justify J PM
RT W
your answer.
SQL
25. R■S = KJRJ 26. KJPJ = KS K
27. PQML = S■Q ■
29. LKWL = M■W TQ
28. P■T = KQ

30. K■P = LQ
KQ

STEM Urban Design  In Washington, D.C., E. Capitol Street, 600 ftLincoln Park
Independence Avenue, C Street, and D Street are
parallel streets that intersect Kentucky Avenue and 800 ft hE sCmap1ito1lgSmt. Msaess_ac0hu7se0tt5s A_AvetS.0t. 5415Kentuc1k0y0A0vfet.14th St.
12th Street. 12th St.
Independence Ave.
31. How long (to the nearest foot) is Kentucky
Avenue between C Street and D Street? Walter St.

32. How long (to the nearest foot) is Kentucky South CarolinCa ASvet..
Avenue between E. Capitol Street and
Independence Avenue?

460 ft 13th St.

Algebra Solve for x. 34. 10x Ϫ 4 D St.
33. 6x
7x 03750. 5_a01222192x ϩ 2
4x 4x ϩ 8
5x 5x Ϫ 1
5x 6x Ϫ 10 7x

36. Think About a Plan  The perimeter of the triangular lot at the right is 50 m.

The surveyor’s tape bisects an angle.hFsimnd1t1hge mlensget_h0s7x0a5n_dty0.5417 hsm11gxmse_0705_yt05418

• How can you use the perimeter to write an equation in x and y?
•h sWmh1a1tgomthesrer_e0la7t0io5n_sth0ip54do16you know between x and y?

37. Prove the Converse of the Side-Splitter Theorem: If a line divides two 12 m 8 m

Proof sides of a triangle proportionally, then it is parallel to the third side.

Given:  RXQR = SYQS Q S
Prove:  RS } XY R2 Y

1
X

hsm11gmse_0705_t05420

476 Chapter 7  Similarity

Determine whether the red segments are parallel. Explain each answer. You can
use the theorem proved in Exercise 37.

38. 6 9 39. 10 40. 12
24
10 15 15
20
12 28 16

41. An angle bisector of a triangle divides the opposite side of the triangle into
segments 5 cm and 3 cm long. A second side of the triangle is 7.5 cm long. Find all

hpossmsi1b1leglemnsgeth_s0f7or0t5h_ett0h5ir4d1s9ide ofhthsemtr1ia1nggmle.se_0705_t05421 hsm11gmse_0705_t05422

42. Open-Ended  In a triangle, the bisector of an angle divides the opposite side
into two segments with lengths 6 cm and 9 cm. How long could the other two
sides of the triangle be? (Hint: Make sure the three sides satisfy the Triangle
Inequality Theorem.)

43. Reasoning  In △ABC, the bisector of ∠C bisects the opposite side. What type of
triangle is △ABC? Explain your reasoning.

Algebra Solve for x. S 45. E9 D
44. Q 2 R 3 A 7.2 F
6
x
3T 7.8 B x C

P

Pr4o6o.f Pphb arsaogmsvepe 14ot71hin3ge,t mdCPro.sarewo_ltl0ah7rey0at5uo_xtihtl0iea5rSy4id2lien4-eSp<CliWtt>earnTdhelaobreeml it.sIninttheehrssdemicatgi1or1anmgwmiftrhsoemli_n0e705_tab0542B3A P W
Given:  a } b } c X
Prove:  BACB = WXYX cC
Y

Pr4o7o .f GdPrriaovwveen t:ht he<AeaDTu>rxbiiailsinaegrclyetsl-iA∠nneCg<BAleEB->B.soisethcatot r<BTEh> e}oDreAm. E. xIntetnhdeCdAiagtorammeefrto<BmE>paatgpeo4in7t3F, . E
Prove:  CDDB = CBAA F

hsm11gmsAe_07305_t05425

C Challenge 48. Use the definition in part (a) to prove the statements in parts (b) and (c). 12 4
a. Write a definition for a midsegment of a parallelogram.
b. A parallelogram midsegment is parallel to two sides of the parallelogram. B

C D

c. A parallelogram midsegment bisects the diagonals of a parallelogram.

hsm11gmse_0705_t05426

Lesson 7-5  Proportions in Triangles 477

49. State the converse of the Triangle-Angle-Bisector Theorem. Give a convincing
argument that the converse is true or a counterexample to prove that it is false.

50. In △ABC, the bisectors of ∠A, ∠B, and ∠C cut the opposite sides into lengths a1
and a2, b1 and b2, and c1 and c2, respectively, labeled in order counterclockwise
around △ABC. Find the perimeter of △ABC for each set of values.

a. b1 = 16, b2 = 20, c1 = 18 b. a1 = 35, a2 = 10 , b1 = 15
3 4

Standardized Test Prep

SAT/ACT 51. What is the value of x in the figure at the right? 12 x
52. Suppose △VLQ ∼ △PSX . If m∠V = 48 and m∠L = 80, 30
2x ϩ 10
what is m∠X ?

53. In the diagram at the right, PR ≅ QR. For what value of x is TS parallel to QP? Q

54. Leah is playing basketball on an outdoor basketball court. The 10-ft pole hsm11gmse_0x Њ70T 5_t05427
supporting the basketball net casts a 15-ft shadow. At the same time, the

length of Leah’s shadow is 8 ft 3 in. What is Leah’s height in inches? You

can assume both Leah and the pole supporting the net are perpendicular PS 56Њ
R
to the ground.

Mixed Review

Use the figure to complete each proportion. a hsm11gSmeesLee_s0s7o0n57_-4t.05428
n
55. nh = ■h 56. ■ = b h c
57. na = ■a b c bm

58. m = ■
h n

Find the center of the circle that you can circumscribe about each △ABC. See Lesson 5-3.

59. A(0, 0) 60. A(2, 5) 61. A( -2, 0)

B(6, 0) B( -2, 5-)1 ) hsm11gmse_0705 _BCt((05-,5254,)25)9
C(0, -6) C( -2,

Get Ready!  To prepare for Lesson 8-1, do Exercises 62–64.

Square the lengths of the sides of each triangle. See p. 829.

62. A 63. A 13 in. B 64. A 4Ί2๵ m
3m 5m 5 in. 12 in. 4m 4m B
C 4m B C
C

hsm11gmse_0705_t05430 hsm11gmse_0705_t05431

478 Chapter 7  Similarity hsm11gmse_0705_t05432

7 Pull It All Together

RMANCPERFOE TASKCompleting the Performance Task

To solve these Look back at your results from the Apply What You’ve Learned sections in Lessons 7-1
problems, you and 7-2. Use the work you did to complete the following.
will pull together
many concepts 1. Solve the problem in the Task Description on page 431 by determining the values
and skills that Lillian should enter for Xmin and Xmax so that the graph is not distorted. Show all
you have studied your work and explain each step of your solution.
about similarity.
2. Reflect  Choose one of the Mathematical Practices below and explain how you
applied it in your work.
MP 1: Make sense of problems and persevere in solving them.

MP 3: Construct viable arguments and critique the reasoning of others.

On Your Own

Lillian graphs the function y = x on her brother’s graphing calculator. She knows that
the slope of the line is 1 and that the line should bisect two of the right angles formed by
the two axes. However, the line does not appear to bisect the right angles on the screen.

Plot1 Plot2 Plot3 WINDOW
\Y1 = X Xmin = –10
\Y2 = Xmax = 10  
\Y3 = Xscl = 1
\Y4 = Ymin = –10
\Y5 = Ymax = 10
\Y6 = Yscl = 1
\Y7 = Xres = 1

 

This time, Lillian wants to adjust the values of Ymin and Ymax so that the graph of the
line is not distorted. In the calculator’s manual, Lillian reads that the screen is 96 pixels
wide by 64 pixels high.

Determine the values Lillian should enter for Ymin and Ymax Lillian so that the graph
is not distorted.

Chapter 7  Pull It All Together 479

7 Chapter Review

Connecting and Answering the Essential Questions

1 Similarity Ratios and Proportions (Lesson 7-1) Proportions in Triangles
You can set up and (Lessons 7-4 and 7-5)
solve proportions using The Cross Products Property states that if
corresponding sides of a c Geometric Means in Right Triangles
similar polygons. b = d , then ad = bc.

2 Reasoning Similar Polygons (Lesson 7-2) ba c
and Proof Corresponding angles of similar polygons
Two triangles are similar are congruent, and corresponding sides of —ea ϭ —af ef —dc ϭ —fc
if certain relationships similar polygons are proportional. d
exist between two
or three pairs of Proving Triangles Similar (Lesson 7-3) —db ϭ —be
corresponding parts. Angle-Angle Similarity (AA ∼) Postulate
Side-Angle-Side Similarity (SAS ∼) Theorem Side-Splitter Theorem
3 Visualization Side-Side-Side Similarity (SSS ∼) Theorem ac
Sketch and label triangles
separately in the same hbsm11gmse_0d 7cr_t0584ab6.ϭaidc
orientation to see how
the vertices correspond. Seeing Similar Triangles E Triangle-Angle-Bisector Theorem
(Lessons 7-3 and 7-4)
hsmc 11gmsed_07cr_t05ab8ϭ47dc.ai
AA
E ab

BD C B CC D
△ABC ∼ △ECD

Chapter Vocabulary hsm11gmse_07cr_t05848.ai

• extended proportion (p. 440) hsm11•g imndsiere_c0t7mcer_astu0r5e8m4e5nt (p. 454) • scale drawing (p. 443)
• extended ratio (p. 433) • means (p. 434) • scale factor (p. 440)
• similar figures (p. 440)
• extremes (p. 434) • proportion (p. 434) • similar polygons (p. 440)

• geometric mean (p. 462) • ratio (p. 432)

Choose the correct term to complete each sentence.

1. Two polygons are ? if their corresponding angles are congruent and
corresponding sides are proportional.

2. A(n) ? is a statement that two ratios are equal.

3. The ratio of the lengths of corresponding sides of two similar polygons is the ? .

4. The Cross Products Property states that the product of the ? is equal to the
product of the ? .

480 Chapter 7  Chapter Review

7-1  Ratios and Proportions

Quick Review Exercises

A ratio is a comparison of two quantities by division. A 5. A high school has 16 math teachers for 1856
math students. What is the ratio of math teachers
proportion is a statement that two ratios are equal. The to math students?
a c
Cross Products Property states that if b = d , where b ≠ 0 6. The measures of two complementary angles are
and d ≠ 0, then ad = bc. in the ratio 2 ∶ 3. What is the measure of the
smaller angle?
Example x 4
∙ 6
What is the solution of x 3 ∙ ? Algebra  Solve each proportion.

6x = 4(x + 3) Cross Products Property 7. 7x = 1281 8. 6 = 15
11 2x
6x = 4x + 12 Distributive Property

2x = 12 Subtract 4x from each side. 9. 3x = x + 4 10. x 8 9 = x 2 3
5 + -
x=6 Divide each side by 2.

7-2 and 7-3  Similar Polygons and Proving Triangles Similar

Quick Review Exercises

Similar polygons have congruent corresponding angles The polygons are similar. Write a similarity statement and
and proportional corresponding sides. You can prove give the scale factor.
triangles similar with limited information about congruent
corresponding angles and proportional corresponding 11. K 28 L 12. Q 12 RX
sides. EH
24 18 24
Postulate or Theorem What You Need 9 6

Angle-Angle (AA ∼) two pairs of ≅ angles J 36 N 12 P P Z 8Y

Side-Angle-Side (SAS ∼) two pairs of proportional sides 13. City Planning  The length of a rectangular
and the included angles ≅
playground in a scale drawing is 12 in. If the scale is
Side-Side-Side (SSS ∼) three pairs of proportional sides
h1s imn. 1=1 1g0m ft,sweh_a0t7isctrh_eta0c5tu8a5lh0les.namgit1h1? gmse_07cr_t05851.a

Example 14. Indirect Measurement  A 3-ft vertical post casts a
24-in. shadow at the same time a pine tree casts a
Is △ABC similar to △RQP? How do B P2 R 30-ft shadow. How tall is the pine tree?
you know? 4

You know that ∠A ≅ ∠R. 8Q Are the triangles similar? How do you know?

AB = AC = 2 , so the triangles A4C 15. A 8Y 16. R P
RQ RP 1 6 G
M C 3E S
are similar by the SAS ∼ Theorem.
10 5 4
T
D

hsm11gmse_07cr_t05849

hsm11gmse_07cr_t0585h2sm.a1i 1gmse_07cr_t05853.ai
Chapter 7  Chapter Review 481

7-4  Similarity in Right Triangles

Quick Review Exercises

CD is the altitude to the C Find the geometric mean of each pair of numbers.
hypotenuse of right △ABC. AD
17. 9 and 16 18. 5 and 12
• △ABC ∙ △ACD,
△ABC ∙ △CBD, and B Algebra  Find the value of each variable. Write your
△ACD ∙ △CBD answer in simplest radical form.

• AD = CD , AB = AADC , and AB = CB 19. 12 6 20.
CD DB AC CB DB

Example hsmll_gmse_07cr_t05854.ai y x x
57
5
What is the value of x?
x 21. 22. 10
5+x = 10 Write a proportion. 10 y
10 5 hsmx 11gymse8_07cr_t0585h6.saymi 11g8mse_07cr_t05857.ai

5(5 + x) = 100 Cross Products Property

25 + 5x = 100 Distributive Property 14 x

5x = 75 Subtract 25 from each side.
x = 15 Divide each side by 5. hsm11gmse_07cr_t05855.ai

7-5  Proportions in Triangles hsm11gmse_07cr_t058h5s9m.ai11gmse_07cr_t05858.ai

Quick Review Exercises

Side-Splitter Theorem and Corollary Algebra  Find the value of x.
If a line parallel to one side of a triangle intersects the other
two sides, then it divides those sides proportionally. If three 23. x 24. x
parallel lines intersect two transversals, then the segments 7 3 6
intercepted on the transversals are proportional. 15 5

Triangle-Angle-Bisector Theorem 14
If a ray bisects an angle of a triangle, then it divides the
opposite side into two segments that are proportional to the 25. 20 8 26. 12 x Ϫ 3
other two sides of the triangle.

Example hsm11xgmse_07cr_t05861hs1m6 11gmse_07xcr_t05862

What is the value of x? 12 9 9
15 x
12 = 9 Write a proportion. 27. 4 28.
15 x
10 63
12x = 135 Cross Products Property 7 x

x = 11.25 Divide each side by 12. hsm11gx mse_07cr_t0586h3sm4511gm55 se_07cr_t05864

482 Chapter 7  Chapter Review hsm11gmse_07cr_t05860
hsm11gmse_07cr_t05865
hsm11gmse_07cr_t05866

7 Chapter Test athXM

MathXL® for SchoolOL
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? 9. Indirect Measurement  A meterstick perpendicular
to the ground casts a 1.5-m shadow. At the same
Algebra  Solve each proportion. time, a telephone pole casts a shadow that is 9 m.
How tall is the telephone pole?
1. 3x = 182 2. x 4 2 = 16
+ 9 10. Photography  A photographic negative is 3 cm
by 2 cm. A similar print from the negative is 9 cm
3. Are the polygons below similar? If they are, write a long on its shorter side. What is the length of the
similarity statement and give the scale factor. longer side?

B E 20 11. What is the geometric mean of 10 and 15?

15 12 16

A 18 C D 24 F Algebra  Find the value of x.

Algebra  The figures in each pair are similar. Find the 12. 6 xϪ6 13. 6
value of each variable. x
10 12

4. 8hsymЊ 11gmse_07ct_t605488 20

8 12 z 14. 15. 8
6
42Њ xЊ x
6
hs1m1 11gmse10_07ct_t0550h0smx 11gmse_0172ct_t05501
5. 2 8

4x Do you UNDERSTAND?

hsm11gmse_07ct_t05491

6. 63Њ y 16. Reasoning  In the diagram, MN } LK . J

xЊ IhssMJmML1e1qguaml tsoeM_LK0N7?cEtx_ptl0ai5n5. 03hsm11gmse_07ct_t05504

hsm11gm4se_07ct_t05493 17. Reasoning  ▱ABCD ∼ ▱PQRS. DB M N

8 16 is a diagonal of ▱ABCD and SQ is a
diagonal of ▱PQRS. Is △BCD similar
Determine whether the triangles are similar. If so, write to △QRS? Justify your reasoning. L K

a similarity statement and name the postulate or the

theorem you used. If not, explain. Determine whether each statement is always,

7. hsm11gmse_07cTt_t05494 sometimes, or never true. hsm11gmse_07ct_t0
P
12 15 18. A parallelogram is similar to a trapezoid.
8 10

R 10 Q W 15 V 19. Two rectangles are similar.

8. G L M 20. If the vertex angles of two isosceles triangles are
congruent, then the triangles are similar.
h12sm710Њ1g1m6 se_07c1t_8 t05740Њ9514

I FN

Chapter 7  Chapter Test 483

7 Common Core Cumulative ASSESSMENT
Standards Review

Some test questions ask you to In the figure below, ABCDE is a regular TIP 2
find the measure of an interior pentagon. What is the measure, in
or exterior angle of a polygon. degrees, of ∠ABE? Find mjA and then use it to
Read the sample question at find mjABE.
the right. Then follow the tips to A
answer it. Think It Through
EB
TIP 1 By the Polygon Angle-Sum

List what you know about Theorem, the sum of the interior
△ABE.
• AE @ AB because the angle measures of ABCDE is
pentagon is regular.
• jABE @ jAEB because (5 - 2)180 = 3(180) = 540.
△ABE is isosceles. 540
• mjA ∙ mjABE ∙ D C So m∠A = 5 = 108. Then
mjAEB ∙ 180 36 72
54 108 108 + m∠ABE + m∠AEB = 180.
#Since ∠ABE ≅ ∠AEB,
-
108 + 2 m∠ABE = 180. So2
m∠ABE = 180 108 = 72 = 36.
2
hsm11gmse_07cu_t05655 The correct answer is A.

VLVeooscsacoabnubluarlayry Builder Selected Response

As you solve test items, you must understand Read each question. Then write the letter of the correct
the meanings of mathematical terms. Match answer on your paper.
each term with its mathematical meaning.
1. What is a name for the quadrilateral below?
A . corollary I. the ratio of a length in a scale
I. square
B . geometric mean drawing to the actual length
II. rectangle
II. a segment connecting the
C. midsegment midpoints of two sides of a III. rhombus

D. scale triangle IV. parallelogram

III. a statement that follows I only II and IV
immediately from a theorem
IV only I, II, and IV

IV. for positive numbers a and b, 2. In which point do the bisectohrssomf t1h1egamngslees_o0f7acu_t05656

the positive number x such triangle meet?
a x
that x = b centroid incenter
circumcenter orthocenter

484 Chapter 7  Common Core Cumulative Standards Review

3. Which quadrilateral does NOT always have 7. Which angle is congruent to ∠DCB?
perpendicular diagonals? C

square kite
rhombus isosceles trapezoid

4. Which of the following facts would be sufficient to AD B
prove △ACE ∙ △BCD?
∠B ∠A
A ∠CDB ∠ACD
B

C 8. hInstmhe1f1iggumresbee_lo0w7,cEuF_ti0s 5a 6m5i9dsegment of △ABC

and ADGC is a rectangle. What is the area of △EDA?
B
49 cm2

D 98 cm2 DE F G
E
294 cm2 14 cm

△BCD is a right triangle. 588 cm2 A 42 cm C
AB ≅ ED
9. Andrew is looking at a map that uses the scale
hsm m11∠gAm=sem_∠07Ecu_t05657
1 in. = 5 mi. On the map, the distance from Westville
AE } BD
to Allentown is 9 in. Which proportion CANNOT be
used to find the actual distanche?sm11gmse_07cu_t05660
5. What is the midpoint of the segment whose endpoints 1 in. 9 din. d 5 mi
are M(6, -11) and N( -18, 7)? 5 mi = 9 in. = 1 in.

( -6, -2) ( -12, 9) 5 mi = 9 iinn.. 5 mi = d
(6, 2) (12, -9) d 1 1 in. 9 in.

10. What type of construction is shown below?

6. Use the figure below. By which theorem or postulate
does x = 3?
X


x2 AB
Y

12 8 angle bisector

SAS Postulate perpendicular bisector

If three parallel lines intersect two transversals, h sm11g cmonsegr_u0e7nctuan_gt0le5s661

hsmat1hr1eegnpmrtohpseeos_retg0iom7nceaunl.t_sti0n5te6r5ce8pted on the transversals congruent triangles

Opposite sides of a parallelogram are congruent. 11. A student is sketching an 11-sided regular polygon.
What is the sum of the measures of the polygon’s first
If two lines are parallel to the same line, then they five angles to the nearest degree?
are parallel to each other.
147 736
720 1620

Chapter 7  Common Core Cumulative Standards Review 485

Constructed Response 19. In rectangle ABCD, AC = 5(x - 2) and 
BD = 3(x + 2). What is the value of x? Justify
12. Triangle ABC is similar to triangle HIJ. Find the area of your answer.
rectangle HIJK.
20. Draw line m with point A on it. Construct a line
H (x) I perpendicular to m at A. What steps did you take to
perform the construction?
C 17 (y)
53 J 2 1. Petra visited the Empire State Building, which is
K approximately 1454 ft tall. She estimates that the scale
A4B of the model she bought is 1 in. = 12 ft. Is this scale
reasonable? Explain.
13. What is the value of x in the figure below?

(4x  15) Extended Response

geom12_se_ccs_c07csr_t01.ai

45 2 2. In the diagram, AB = FE, CD

14. In hexagon ABCDEF, ∠A and ∠B are right angles. BC = ED, and AE = FB. G
If ∠C ≅ ∠D ≅ ∠E ≅ ∠F , what is the measure of ∠F a. Is there enough BE
in degrees?
information to prove
15. A shcaslme d1r1agwminsgeo_f 0a7swcuim_tm0i5n6g6p2ool and deck is shown
△BCG ≅ △EDG? Explain.
below. Use the scale 1 in. = 2 m. What is the area of
the deck in square meters? b. What one additional

piece of information AF

would allow you to prove

△BCD ≅ △EDC? Explain.

c. What can you conclude from the diagram that

2 in. 4 in. would help you prove △BAF ≅bhy△sSmAESF1Ao1r?gSmSSs?e_07cu_t05665
d. In part (c), is △BAF ≅ △EFA

Explain.

6 in. 2 3. At a campground, the 50-yd path from your campsite
8 in. to the information center forms a right angle with
the path from the information center to the lake.
16. In parallelogram ABCD below, DB is 15. What is DE? The information center is located 30 yd from the
bathhouse. How far is your campsite from the lake?
A B Show your work.

hsm11gmse_0E 7cu_t05663

Information center

DC 50 yd 30 yd

17. The measure of the vertex angle of an isosceles triangle
is 112. What is the measure of a base angle?

18. hWshmat1is1tghme vsael_u0e7ocf ux _int0th5e6f6ig4ure below?

5 Your campsite x Bathhouse y Lake

x

43

hsm11gmse_07cu_t05666

486 hsCmha1p1tegrm7 seC_o0m7mcuon_tC0o5re65Cu4mulative Standards Review