Geometry

9-6 Dilations CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To understand dilation images of figures MG-ASFRST.9A1.12a.G  -ASdRiTla.1ti.o1nat aAkedsilatliionne tnaoktepsaassliinnegnthortough
pthaesscienngtethrroofutghhetdhielactieonntetrooaf pthaeradllielaltlionne,t.o. a. Aplasroallel
lGin-eS,R. T...AA.1lsbo, GM-ACFOS.A.9.122, .GG--SSRRTT..A1.21b,
MMAPF1S,.M91P2.3G, -MCOP.41,.2M, MP 7AFS.912.G-SRT.1.2
MP 1, MP 3, MP 4, MP 7

The pupil is the opening in the iris that lets light into the eye.
Depending on the amount of light available, the size of the pupil
changes.

Do you think Normal Light Dim Light
you can model
this using rigid Iris Pupil
motions?
12 mm 12 mm
MATHEMATICAL Diameter of pupil = 2 mm Diameter of pupil = 8 mm

PRACTICES

Observe the size and shape of the iris in normal light and in dim light.
What characteristics stay the same and what characteristics change?
How do these observations compare to transformation of figures you
have learned about earlier in the chapter?

Lesson In the Solve It, you looked at how the pupil of an eye changes in size, or dilates. In this
lesson, you will learn how to dilate geometric figures.
Vocabulary
• dilation Essential Understanding  You can use a scale factor to make a larger or smaller
• center of dilation copy of a figure that is also similar to the original figure.
• scale factor of a
Key Concept  Dilation
dilation
• enlargement
• reduction

A dilation with center of dilation C and scale PЈ
RЈ
factor n, n 7 0, can be written as D(n, C). A dilation
is a transformation with the following properties.
P
#• The image of C is itself (that is, CCR′ >=anCd). Q QЈ
For any other point R, R′ is on 587
• CR ′ R
CR′ = n CR, or n = CR . CЈ ϭ C CRЈ ϭSn и CR

• Dilations preserve angle measure.

Lesson 9-6  Dilations

The scale factor n of a dilation is the ratio of a length of the image to the corresponding

length in the preimage, with the image length always in the numerator. For the figure

shown on page 587, n = CR′ = R′P′ = P′Q′ = QQ′RR ′ .
CR RP PQ

A dilation is an enlargement if the scale factor n is greater than 1. The dilation is a

reduction if the scale factor n is between 0 and 1.

BЈ CЈ F G

BC 4 FЈ C GЈ
EЈ HЈ
2 2

AЈ ϭ A D DЈ E 8H

Enlargement Reduction 1
4
center A, scale factor 2 center C, scale factor

Problem 1 Finding a Scale Factor

Mhuslmtip1l1egCmhosiece_ 0I9s 0D5(n_,tX0)8(△14X0T.Ra)i = △X′T′R′ an enlargement RЈ TЈ

or a reduction? What is the scale factor n of the dilation?

enlargement; n = 2 reduction; n = 1 R 8
enlargement; n = 3 3 4
Wf ahctyoirs the scale T
not 142, or reduction; n = 3 XЈ ϭ X
1 ?
3
The image is larger than the preimage, so the dilation is an
The scale factor of a
enlargement.
dilation always has the

image length (or the Use the ratio of the lengths of corresponding sides to find the scale factor.

distance between a point X′T ′ 4 + 8 12
XT 4 4
on the image and the n = = = = 3 hsm11gmse_0905_t08144.ai

center of dilation) in the △X ′T′R′ is an enlargement of △XTR, with a scale factor of 3. The correct answer is B.

numerator.

Got It? 1. Is D(n, O) (JKLM) = J′K′L′M′an enlargement or a yJ
reduction? What is the scale factor n of the dilation?
JЈ K
x
Ϫ2 O 2 KЈ4 L

M MЈ LЈ
Ϫ3

In Got It 1, you looked at a dilation of a figure drawn in the hsm1y 1gmse_0905_t08145.ai
coordinate plane. In this book, all dilations of figures in the
coordinate plane have the origin as the center of dilation. So ny P؅(nx, ny)
you can find the dilation image of a point P(x, y) by multiplying
the coordinates of P by the scale factor n. A dilation of scale factor y P(x, y) OP؅ ‫ ؍‬n ؒ OP
n with center of dilation at the origin can be written as
Ox x
Dn (x, y) = (nx, ny) nx

588 Chapter 9  Transformations hsm11gmse_0905_t08148.ai

Will the vertices of Problem 2 Finding a Dilation Image 2y x
the triangle move Z
closer to (0, 0) or What are the coordinates of the vertices of
farther from (0, 0)? D2(△PZG)? Graph the image of △PZG. Ϫ6 Ϫ4 Ϫ2 O P4

The scale factor is 2, Identify the coordinates of each vertex. The center of G
dilation is the origin and the scale factor is 2, so use the
so the dilation is an dilation rule D2(x, y) = (2x, 2y). Z؅ Z 2 y
Ϫge6omϪ142_se_ccsO_c09l0P6_t043P.؅aix
enlargement. The vertices # # D2(P) = (2 2, 2 (-1)), or P′(4, -2).
# # D2(Z) = (2 (-2), 2 1), or Z′(-4, 2). G
will move farther from # #D2(G) = (2 0, 2 (-2)), or G′(0, -4). G؅

(0, 0). To graph the image of △PZG, graph P′, Z′, and G′.
Then draw △P′Z′G′.

Got It? 2. a. What are the coordinates of the vertices of D1 (△PZG)?
2

b. Reasoning  How are PZ and P′Z′related? How are PG and P′G′, and GZ

and G′Z′related? Use these relationships to makgeeaocmon1je2c_tsuere_acbcos_utct0h9el06_t04.ai
effects of dilations on lines.

Dilations and scale factors help you understand real-world enlargements and
reductions, such as images seen through a microscope or on a computer
screen.

Problem 3 Using a Scale Factor to Find a Length

What does a scale Biology  A magnifying glass shows you an image of an object
that is 7 times the object’s actual size. So the scale factor of the
factor of 7 tell you? enlargement is 7. The photo shows an apple seed under this

A scale factor of 7 tells # #magnifying glass. What is the actual length of the apple seed?
1.75 = 7 p image length = scale factor actual length
you that the ratio of

the image length to the

actual length is 7, or

image length = 7. 0.25 = p Divide each side by 7.
actual length

The actual length of the apple seed is 0.25 in.

Got It? 3. The height of a document on your computer 1.75 in.
screen is 20.4 cm. When you change the zoom
setting on your screen from 100% to 25%, the
new image of your document is a dilation of the
previous image with scale factor 0.25. What is
the height of the new image?

Lesson 9-6  Dilations 589

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

1. The blue figure is a dilation 12 5. Vocabulary  Describe the scale factor of a reduction.
image of the black figure with 8
center of dilation C. Is the 6. Error Analysis  The blue 2 A1
dilation an enlargement or a C figure is a dilation image 63
reduction? What is the scale of the black figure for a
factor of the dilation? dilation with center A.

Find the image of each point. hsm11gmse_0905_t09439.ai Two students made errors when asked to find the
scale factor. Explain and correct their errors.

2. D2 (1, - 5) 3. D1 (0, 6) 4. D10 (0, 0) A. B.
2 hsm1n1g=m41se=_40905_t09440.ai
n= 2 = 1
6 3

Practice and Problem-Solving Exercises hsmPM1RA1TAHgCEmMTAsITCeIC_EA0SL905_t09441h.sami 11gmse_0905_t09442.ai

A Practice The blue figure is a dilation image of the black figure. The labeled point is the See Problem 1.
center of dilation. Tell whether the dilation is an enlargement or a reduction.
Then find the scale factor of the dilation. 2

7. 8. 9.

4 C4
A6 3

R
9

10. 11. 12.

hsm11gmse_0905_t06790.ai

hsm11gmse_0905_t06791.ai hsm11gmse_0905_t06792.ai

KL M

13. y 14. y 15. y

6 hsm11gm1se_0905_x t06794.ai 6
h4sm11gmse_0905_t06793.ai Ϫ2 O 3 hsm11gmse_09405_t06795.ai

2

x x
O 246 Ϫ6 Ϫ4 Ϫ2 O

590 Chapter 9  Transformations hsm11gmse_0905_t06797.ai hsm11gmse_0905_t06798.ai
hsm11gmse_0905_t06796.ai

Find the images of the vertices of △PQR for each dilation. Graph the image. See Problem 2.

16. D3 (△PQR) 17. D10 (△PQR) 18. D3 (△PQR)
4y Q 4
Qy Qy

2 x P 2 1x
O1 5 Ϫ4 P O2
Ϫ2 P R Ox
1 Ϫ3 R

Ϫ3 R

Magnification  You look at each object described in Exercises 19–22 under a See Problem 3.

magnifying glass. Find the actual dimension of each object.

19. Thhsemi1m1gamgeseo_f0a90b5u_ttt0o6n7i9s95.atiimes the bhustmto1n1g’smascet_u0a9l0s5iz_et0a6n8d00h.aais a diamethesrmo1f16gcmmse. _0905_t06801.ai

20. The image of a pinhead is 8 times the pinhead’s actual size and has a width of
1.36 cm.

21. The image of an ant is 7 times the ant’s actual size and has a length of 1.4 cm.

22. The image of a capital letter N is 6 times the letter’s actual size and has a height of
1.68 cm.

B Apply Find the image of each point for the given scale factor.

23. L( -3, 0); D5 (L) 24. N( - 4, 7); D0.2 (N) 25. A( - 6, 2); D1.5 (A)

26. F(3, - 2); D1 (F) ( )27. B45,- 3 ; D 1 (B) ( )28. Q6, 13 ; D16 (Q)
3 2 10 2

Use the graph at the right. Find the vertices of the image of QRTW for a yW
dilation with center (0, 0) and the given scale factor. Q4

29. 14 30. 0.6 31. 0.9 32. 10 33. 100 2
T

34. Compare and Contrast  Compare the definition of scale factor of a dilation Ϫ3 Ϫ1 O 2 x
to the definition of scale factor of two similar polygons. How are they alike? R
How are they different?

35. Think About a Plan  The diagram at the right shows △LMN and 2 4hsmL 11xgmLЈse_0x9ϩ053_t0M68Ј 02.ai
its image △L′M′N′ for a dilation with center P. Find the values of M
P yЊ N
x and y. Explain your reasoning. NЈ(2y Ϫ 60)Њ
• What is the relationship between △LMN and △L′M′N′?
• What is the scale factor of the dilation?

• Which variable can you find using the scale factor?

36. Writing  An equilateral triangle has 4-in. sides. Describe its image for

a dilation with center at one of the triangle’s vertices and scale factor 2.5.

hsm11gmse_0905_t06804.ai

Lesson 9-6  Dilations 591

Coordinate Geometry  Graph MNPQ and its image M′N′P′Q′ for a dilation with
center (0, 0) and the given scale factor.

37. M(1, 3), N( -3, 3), P( -5, -3), Q( -1, -3); 3 38. M(2, 6), N( -4, 10), P( -4, - 8), Q( - 2, - 12); 1
4

39. Open-Ended  Use the dilation command in geometry software or

drawing software to create a design that involves repeated dilations,
such as the one shown at the right. The software will prompt you to

specify a center of dilation and a scale factor. Print your design and
color it. Feel free to use other transformations along with dilations.

A dilation maps △HIJ onto △H′I′J′. Find the missing values.

40. HI = 8 in.  H′I′ = 16 in. 41. HI = ■ ft  H′I′ = 8 ft

IJ = 5 in.   I′J′ = ■ in. IJ = 30 ft  I′J′ = ■ ft

HJ = 6 in.  H′J′ = ■ in. HJ = 24 ft  H′J′ = 6 ft

42. Let / be a line through the origin. Show that Dk(/) = / by showing that if C = (c1, c2)
is on /, then Dk(C) is also on /.
hsm11gmse_0905_t09542.ai
43. Lsuept pAo=se(tah1a,ta<A2)Ba> nddoeBs = (b1, b2), let A′ = Dk(A) and B′ = Dk(B) with k ≠
not pass through the origin. 1, and

a. Show that <AB> ≠ <A′B′> (Hint: What happens to the x- and y-intercepts of <AB>

under the dilation Dk?) that <AB> is parallel to <A ′B′> by showing that they
b. Suppose that a1 ≠ b1. Show
have the same slope.
c. Show that <AB> ͉ ͉ <A′B′> if a1 = b1.

44. Reasoning  You are given AB and its dilation image A′B′ with A, B, A′, and B′
noncollinear. Explain how to find the center of dilation and scale factor.

Reasoning  Write true or false for Exercises 45–48. Explain your answers.
45. A dilation is an isometry.
46. A dilation with a scale factor greater than 1 is a reduction.
47. For a dilation, corresponding angles of the image and preimage are congruent.
48. A dilation image cannot have any points in common with its preimage.

C Challenge Coordinate Geometry  In the coordinate plane, you can extend dilations to

include scale factors that are negative numbers. For Exercises 49 and 50, use
△PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).

49. Graph D-3 (△PQR).

50. a. Graph D-1 (△PQR).
b. Explain why the dilation in part (a) may be called a reflection through a point.

Extend your explanation to a new definition of point symmetry.

592 Chapter 9  Transformations

51. Shadows  A flashlight projects an image B BЈ CЈ
C AЈ DЈ
of rectangle ABCD on a wall so that each
AD
vertex of ABCD is 3 ft away from the
corresponding vertex of A′B′C′D′. The
length of AB is 3 in. The length of A′B′ is
1 ft. How far from each vertex of ABCD

is the light?

Standardized Test Prep

SAT/ACT 52. A dilation maps △CDE onto △C′D′E′. If CDhs=m171.5gmft,seC_E09=051_5tf0t6, 8D0′7E.a′i= 3.25 ft,
and C′D′ = 2.5 ft, what is DE?

1.08 ft 5 ft 9.75 ft 19 ft

53. You want to prove indirectly that the diagonals of a rectangle are congruent. As the
first step of your proof, what should you assume?

A quadrilateral is not a rectangle.

The diagonals of a rectangle are not congruent.

A quadrilateral has no diagonals.

The diagonals of a rectangle are congruent.

54. Which word can describe a kite?

equilateral equiangular convex scalene

Short 55. U se the figure at the right to answer the questions below.
Response a. Does the figure have rotational symmetry? If so, identify the angle of rotation.

b. Does the figure have reflectional symmetry? If so, how many lines of symmetry

does it have?

Mixed Review SeehsLmes1s1ognm9s-e5_. 0905_t0
See Lesson 7-2.
56. △JKL has vertices J(23, 2), K(4, 1), and L(1, 23). What are the coordinates of
J ′, K ′, and L ′ if (Rx@axis ∘ T62, -37)(△JKL) = △J ′K ′L′? 593

Get Ready!  To prepare for Lesson 9-7, do Exercises 55–57.

Algebra  TRSU ∼ NMYZ. Find the value of each variable. M
4.5 50Њ
57. a 58. b 59. c R N 150Њ Y
3
6b
T 120Њ S cЊ

a5

Z
U

Lesson 9-6  Dilations

9-7 Similarity CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Transformations
MG-ASFRST.9A1.2 .GG-iSveRnT.t1w.o2 fiGgiuvreens,tuwsoe ftihgeurdeesf,inuisteion of
tshimeidlaerfiitnyitiinonteormf simofilsairmitiylairnittyetrrmasnsoffosrmimaitlaiorintsy to decide
tifrathnesfyoarmreastimonilsatro. .d.eAcildseoifGt-hSeRy Ta.rAe.s3imilar . . . Also
MMPAF1S,.M91P2.2G, -MSRPT3.1, .M3 P 4
MP 1, MP 2, MP 3, MP 4

Objectives To identify similarity transformations and verify properties of similarity

Your friend says that she performed A AЈ
BC CЈ
a composition of transformations to
map △ABC to △A′B′C′. Describe the

composition of transformations.

Is there more than BЈ
one composition of
transformations
possible to map
△ABC to △A′B′C′?

MATHEMATICAL geom12_se_ccs_c09l07_t01.Ai

PRACTICES In the Solve It, you used a composition of a rigid motion and a dilation to describe the

mapping from △ABC to △A′B′C′.

Lesson Essential Understanding  You can use compositions of rigid motions and
dilations to help you understand the properties of similarity.
Vocabulary
• similarity

transformation
• similar

Problem 1 Drawing Transformations

△DEF has vertices D(2, 0), E(1, 4), and F(4, 2). 6y
What is the image of △DEF when you apply the EЈ4 E
composition D1.5 ∘ Ry@axis?

Step 1 Find the vertices of Ry@axis(△DEF). Then FЈ 2 F
connect the vertices to draw the image.
Ϫ6 Ϫ4 DЈ O D 4 x
Ry@axis (D) = D′( - 2, 0) 6
 Ry@axis (E ) = E′( - 1, 4)
  Ry@axis (F ) = F′( - 4, 2) x
6
Step 2 Find the vertices of the dilation of △D′E′F′. EЉ 6 y
Then connect the vertices to draw the image.
EЈ4 E
D1.5 (D′) = D″( - 3, 0) FЉ
  D1.5 (E′) = E″( - 1.5, 6) FЈ 2 F
  D1.5 (F′) = F″( - 6, 3)
geo12_se_ccs_c09l07_t02_patches.ai
The vertices of the image after the composition of
transformations are D″( -3, 0), E″( -1.5, 6), and Ϫ6 Ϫ4 DЉ DЈ O D 4

F″( -6, 3).

594 Chapter 9  Transformations

Got It?MATHEMATICAL 1. Reasoning  △LMN has vertices L( -4, 2), M( -3, -3), and N( -1, 1).

PRACTICES

Suppose the triangle is translated 4 units right and 2 units up and then

dilated by a scale factor of 0.5 with center of dilation at the origin. Sketch the

resulting image of the composition of transformations.

MATHEMATICAL 4 y
S
PPRrAoCbTlIeCmES 2 Describing Transformations

What is a composition of rigid motions and a dilation that
maps △RST to △PYZ?

2

The vertices of A composition of Study the figures to RT
the preimage transformations determine how the image Ϫ6 Ϫ4 Ϫ2 O 2 x
and image that maps △RST could have resulted Z PϪ2
to △PYZ from the preimage.
Then use the vertices to Ϫ4
verify the composition
of transformations. YϪ6

It appears that △RST was rotated and then enlarged to create △PYZ. To verify the
composition of transformations, begin by rotating the triangle 180° about the origin.

r(180°, O) (R) = R′( - 1, - 1) Use the rule r(180°, O)(x, y) = ( - x, - y). geo12_se_ccs_c09l07_t04_patch
r(180°, O) (S) = S′( - 1, - 3)
r(180°, O) (T ) = T ′( - 3, - 1)

△PYZ appears to be about twice as large as △RST. Scale the vertices of the
intermediate image R′S′T ′ to verify the composition.

D2( - 1, -1) = P( - 2, - 2) Use the rule D2(x, y) = (2x, 2y).
D2( - 1, -3) = Y( - 2, - 6)
D2( - 3, - 1) = Z( - 6, - 2)

The vertices of the dilation of △R′S′T′ match the vertices of △PYZ.

A rotation of 180° about the origin followed by a dilation with scale factor 2
maps △RST to △PYZ.

Got It? 2. What is a composition of rigid motions y

and a dilation that maps trapezoid ABCD A B 2N M
to trapezoid MNHP?
x

Ϫ8 Ϫ6 Ϫ4 Ϫ2 O 2 6
D CH P

geo12_se_ccs_c09l07_t05_patches.ai

Lesson 9-7  Similarity Transformations 595

Notice that the figures in Problems 1 and 2 appear to have the same shape but different
sizes. Compositions of rigid motions and dilations map preimages to similar images.
For this reason, they are called similarity transformations. Similarity transformations
give you another way to think about similarity.

Key Concept  Similar Figures

Two figures are similar if and only if there is a similarity transformation that maps one figure
onto the other.

Here’s Why It Works  Consider the composition of a Q R
rigid motion and a dilation shown at the right. RЈ
P
Because rigid motions and dilations preserve angle measure, PЈ

m∠P = m∠P′, m∠Q = m∠Q′, and m∠R = m∠R′. So, QЈ
corresponding angles are congruent.

Because there is a dilation, there is some scale factor k such that:

 PQ = kP′Q′  QR = kQ′R′   PR = kP′R′

  k = PQ   k = QQ′RR′  k = PR geom12_se_ccs_c09l07_t06.ai
P′Q′ P′R′

So, PQ = QR = PP′RR′.
P′Q′ Q′R′

Problem 3 Finding Similarity Transformations

Is there a similarity transformation that maps △PAQ P T N
to △TNO? If so, identify the similarity transformation O

and write a similarity statement. If not, explain.

Does it matter #Although PA ≠ TN, there is a scale factor k such that A
what the center of
dilation is? k PA = TN. Dilate △PAQ using this scale factor. Then
P′A′ ≅ TN. Since dilations preserve angle measure,
#No. All that matters is
you also know that ∠P′ ≅ ∠T and ∠A′ ≅ ∠N. Q
that k PA = TN. Therefore, △P′A′Q′ ≅ △TNO by ASA. This means that

there is a sequence of rigid motions that maps △P′A′Q′ onto △TNO.

So, there is a dilation that maps △PAQ to △P′A′Q′, and a sequence of rigid motions

that maps △P′A′Q′ to △TNO. Therefore, there is a composition of a dilation and rigid
geo12_se_ccs_c09l07_patch G.ai
motions that maps △PAQ onto △TNO.

Got It? 3. Is there a similarity transformation that maps R 20
△JKL to △RST ? If so, identify the similarity K T
transformation and write a similarity statement.
10 12 12 16
If not, explain. S
J L
16

596 Chapter 9  Transformations

How can you Similarity transformations provide a powerful general approach to similarity. In
determine whether Problem 3, you used similarity transformations to verify the AA Postulate for triangle
two figures are similarity. Another advantage to the transformational approach to similarity is that you
similar if you have can apply it to figures other than polygons.
no information about
side lengths or angle MATHEMATICAL
measures?
Any two plane figures PPRrAoCbTlIeCmES 4 Determining Similarity
are similar if you
can find a similarity A new company is using a computer program to design its
transformation that maps logo. Are the two figures used in the logo so far similar?
one onto the other.
If you can find a similarity transformation between two
figures, then you know they are similar. The smaller lightning
bolt can be translated so that the tips coincide. Then it can be
enlarged by some scale factor so that the two bolts overlap.

The figures are similar because there is a similarity
transformation that maps one figure onto the other. The
transformation is a translation followed by a dilation.

Got It? 4. Are the figures at the right similar? Explain.

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Use the diagram below for Exercises 1 and 2.
PRACTICES

1. What is a similarity transformation that maps △RST PM3RAT.AH CEVMTAIoTCICEbASLabulary  Describe how the word dilation is
to △JKL? used in areas outside of mathematics. How do these

applications relate the mathematical definition?

2. What are the coordinates of (D1 ∘ r(180°, O))(△RST)? 4. OpenMATHEMATICAL Ended  For △TUV at T y
4 Ϫ6 Ϫ4 V
PRACTICES
U 2
the right, give the vertices
x
y of a similar triangle after a O
6L
similarity transformation

T4 that uses at least 1 rigid

motion.

R2 K J
2
S x
Ϫ6 Ϫ4 Ϫ2 O 46

Ϫ2

Lesson 9-7  Similarity Transformations geom12_se_ccs_5c9079l07_

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice △MAT has vertices M(6, −2), A(4, −5), and T(1, −2). For each of the See Problem 1.
following, sketch the image of the composition of transformations.

5. reflection across the x-axis followed by a dilation by a scale factor of 0.5

6. rotation of 180° about the origin followed by a dilation by a scale factor of 1.5

7. translation 6 units up followed by a reflection across the y-axis and then a dilation
by a scale factor of 2

For each graph, describe the composition of transformations that maps △FGH See Problem 2.

to △QRS.

8. 4 y G 9. 4 y 10. 4 y Q
RS
RF x F2 x
QS 4 4
Ϫ4 Ϫ2 O 2 F x H
Ϫ4 2Q4
Ϫ2 Ϫ4 Ϫ2
H G
Ϫ4 GS
Ϫ4 H Ϫ4 R

For each pair of figures, determine if there is a similarity transformation that See Problem 3.

maps one figure onto the other. If so, identify the similarity transformation

and write a similarity statement. If not, explain.

g1e1o. mA12_9se_ccs_c09l07_t017.ag1ie2o. mA12_s2e0_ccsB_c09l07_t018.a1i3. g eom12_Sse_ccs_c09l07_t019.ai

18 V I 8 C
15 10 DC LA
S G 12
6 R H 12 K E

25

geom12_se_ccs_c09l07_t020.ai IJ

geo12_se_ccs_c09l07_t022.ai

Determine whether or not each pair of figures below is similar. Explain your See Problem 4.

reasoning.

14. 15.
geom12_se_ccs_c09l07_t021.Ai

598 Chapter 9  Transformations

B Apply 16. Writing  Your teacher uses geometry software program to plot △ABC with vertices
A(2, 1), B(6, 1), and C(6, 4). Then he used a similarity transformation to plot △DEF
with vertices D( -4, -2), E( -12, -2), and F( -12, -8). The corresponding angles of
the two triangles are congruent. How can the Distance Formula be used to verify

that the ratios of the corresponding sides are proportional? Verify that the figures

are similar.

17. Think About a Plan  Suppose that △JKL is formed by connecting B
JK
the midpoints of △ABC. Is △AJL similar to △ABC? Explain.
L
• How are the side lengths of △AJL related to the side lengths geo12_se_ccs_c09l07_t0050.ai

of △ABC ?

• Can you find a similarity transformation that maps △AJL to

△ABC ? Explain. A C

18. Writing  What properties are preserved by rigid motions but not

by similarity transformations?

Determine whether each statement is always, sometimes, or never true.

19. There is a similarity transformation between two rectangles.

20. There is a similarity transformation between two squares.

21. There is a similarity transformation between two circles.

22. There is a similarity transformation between a right triangle and an equilateral
triangle.

23. Indirect Measurement  A surveyor wants to 178 m d
use similar triangles to determine the distance 75º 75º
across a lake as shown at the right. 132 m

a. Are the two triangles in the figure similar?
Justify your reasoning.

b. What is the distance d across the lake?

445 m

24. Photography  A 4-inch by 6-inch rectangular
photo is enlarged to fit an 8-inch by 10-inch frame. Are the two photographs
similar? Explain.

25. Reasoning  Is a rigid motion an example of a similarity transformation? Explain
your reasoning and give an example.

26. Art  A printing company enlarges a banner for a 13 in.

graduation party by a scale factor of 8. 3 in.
a. What are the dimensions of the larger banner?
b. How can the printing company be sure that the

enlarged banner is similar to the original?

Lesson 9-7  Similarity Transformations 599

C Challenge 27. If △ABC has vertices given by A(u, v), B(w, x), and C(y, z), and △NOP has vertices
given by N(5u, -4v), O(5w, -4x), and P(5y, -4z), is there a similarity transformation
that maps △ABC to △NOP? Explain.

28. Overhead Projector  When Mrs. Sheldon places a transparency on the screen of the
overhead projector, the projector shows an enlargement of the transparency on the
wall. Does this situation represent a similarity transformation? Explain.

29. Reasoning  Tell whether each statement below is true or false.
a. In order to show that two figures are similar, it is sufficient to show that there is a

similarity transformation that maps one figure to the other.
b. If there is a similarity transformation that maps one figure to another figure, then

the figures are similar.
c. If there is a similarity transformation that maps one figure to another figure, then

the figures are congruent.

Standardized Test Prep

SAT/ACT 30. △STU has vertices S(1, 2), T(0, 5), and U( -8, 0). What is the x-coordinate of S after
a 270° rotation about the origin?

31. The diagonals of rectangle PQRS intersect at O. PO = 2x - 5 and OR = 7 - x.
What is the length of QS?

32. The length of the hypotenuse of a 45°-45°-90° triangle is 55 in. What is the length of
one of its legs to the nearest tenth of an inch?

33. You place a sprinkler so that it is equidistant from three rose bushes at B
points A, B, and C. How many feet is the sprinkler from A?
3 yd
4 yd
A C

Mixed Review hsm11gmse_0904_t14048

34. Which capital letters of the alphabet are rotation images of themselves? See Lesson 9-3.

Draw each letter and give an angle of rotation (6 360°) .

35. Three vertices of an isosceles trapezoid are ( -2, 1), (1, 4), and (4, 4). Find See Lesson 6-7.

all possible coordinates for the fourth vertex.

Get Ready!  To prepare for Lesson 10-1, do Exercises 34–37.

Find the area of each figure. See Lesson 1-8.

36. a square with 5-cm sides 37. a rectangle with base 4 in. and height 7 in.
38. a 4.6 m-by-2.5 m rectangle
39. a rectangle with length 3 ft and width 1 ft
2

600 Chapter 9  Transformations

9 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 9-2,
problems you 9-3, and 9-4. Use the work you did to complete the following.
will pull together
many concepts 1. Solve the problem in the Task Description on page 543 by finding a sequence of
and skills transformations that uses as few moves as possible to move the puzzle piece to the
that you have target area for Case 1 and for Case 2. Show all your work and explain each step of your
studied about solution.
transformations.
2. Reflect  Choose one of the Mathematical Practices below and explain how you applied
it in your work on the Performance Task.
MP 1: Make sense of problems and persevere in solving them.

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

MP 5: Use appropriate tools strategically.

On Your Own

Alicia must write a program that moves the puzzle piece △PQR to the target area △XYZ,
as shown in the graph below. Find a sequence of three different transformations (one
translation, one reflection, and one rotation about the origin) that Alicia can use in her
program.

y

6

4 Z x
2 6
2
Ϫ6 Ϫ4 Ϫ2 O Y X
P Ϫ2

QR
Ϫ2

Chapter 9  Pull It All Together 601

9 Chapter Review

Connecting and Answering the Essential Questions

1 Transformations Transformations Composing Transformations
When you translate, reflect, (Lesson 9-4)
or rotate a geometric figure, (Lessons 9-1, 9-2, 9-3, and 9-6)
its size and shape stay the A glide reflection moves the black
same. When you dilate a The black triangle is the preimage of each triangle down 3 units and then
geometric figure, the figure transformation. reflects it across the line x = -2.
is enlarged or reduced.
4y Dilation 2y
2 Coordinate
Geometry Reflection Ϫ6 Ϫ4 x
You can show a O2
transformation in the x
coordinate plane by Ϫ4 Ϫ2 246 8 Ϫ2
graphing a figure and its Ϫ2 Rotation
image.
Ϫ4 Translation Similarity Transformations
3 Visualization
If two figures are congruent, Congruence Transformations (Lesson 9-7)
then you can visualize a (Lesson 9-5)
congruence transformation Fhigsumres1a1rge msimsiela_r0if9acnrd_otn1ly05if06.ai
that maps one figure to the Tihsrisaamnseg1qleu1seganrmceecsooefn_rgig0rui9decnmrto_ifttiao1nn0ds5toh0na3lyt.maifiathpesre
other. If you can visualize there is a sequence of rigid motions
a composition of rigid one triangle to the other. and dilations that maps one figure
motions and dilations that Because △A″B″C″ is the image of onto the other.
map one figure to another, △ABC after a reflection and a translation, Because △L″M″N″ is the image
then the figures are similar. △ABC ≅ △A″B″C″. of △LMN after a reflection and
a dilation, △LMN is similar to
△L″M″N″.

n LЈ LЉ
MЈ
A AЈ AЉ
M MЉ

B C nCЈ CЉ BЉ L NЈ
BЈ N NЉ

Chapter Vocabulary

• composition of • dilation (p. 587) • preimage (p. 545) geom• (s1pi2m._5isle9a_6ric)tcys_trca0n9csfr_otr0m2a.atiion
transformations (p. 548) • transformation (p. 545)
• glide reflection (p. 572) • reflection (p. 554) • translation (p. 547)
• congruence • imaggeeo(pm.1524_5s)e_ccs_c09cr_t01.•a irigid motion (p. 545)
transformation (p. 580)
• isometry (p. 570) • rotation (p. 561)

Choose the correct term to complete each sentence.

1. A(n) ? is a change in the position, shape, or size of a figure.

2. A(n) ? is a composition of rigid motions and dilations.

3. In a(n) ? , all points of a figure move the same distance in the same direction.

4. A(n) ? is the result of a transformation.

602 Chapter 9  Chapter Review

9-1 Translations Exercises L F
Z O
Quick Review 5. a. A transformation maps
ZOWE onto LFMA. Does E W
A transformation of a geometric figure is a change in its the transformation appear to A M
position, shape, or size. be a rigid motion? Explain.
A translation is a rigid motion that maps all points of a
figure the same distance in the same direction. b. What is the image of ZE?
In a composition of transformations, each What is the preimage of M?
transformation is performed on the image of the preceding
transformation. 6. △RST has vertices R(0, -4), S( -2, -1), and

Example 7. TW(r-it6e,a1r).uGleratophdeTs6c-ri4b, e77a(△traRnSsTla).thiosnm51u1ngitms lesfet_a0n9dcr_t10516.a

What are the coordinates of T6-2, 37(5, - 9)? 10 units up.
Add -2 to the x-coordinate, and 3 to the y-coordinate.
A(5, -9) S (5 - 2, -9 + 3), or A′(3, -6). 8. Find a single translation that has the same effect as
the following composition of translations.
9-2 Reflections T6-4, 77 followed by T63, 07

Quick Review Exercises

The diagram shows a reflection across line r. A reflection Given points A(6, 4), B( −2, 1), and C(5, 0), graph △ABC
is rigid motion that preserves distance and angle measure. and each reflection image.
The image and preimage of a reflection have opposite
orientations. 9. Rx-axis(△ABC)
10. Rx=4(△ABC)
r 11. Ry=x(△ABC)
12. Copy the diagram. Then draw Ry-axis (BGHT). Label
Example
the vertices of the image by using prime notation.

y
4B

Use points P(1, 0), Q(3, −2), and R(4, 0). What is
Ry-axis(△PQR)? hsm11gmse_09cr_t10508.ai
T Gx
Graph △PQR. Find P′, Q′, y Rx Ϫ4 Ϫ2 4
5
and R′ such that the y-axis RЈ PЈ P H
O Ϫ4
is the perpendicular

bisector of PP′, QQ′, and QЈ Ϫ2 Q
RR′. Draw △P′Q′R′.

hsm11gmse_09cr_t10511.ai 603

geom12_se_ccs_c09cr_t03.ai
Chapter 9  Chapter Review

9-3 Rotations

Quick Review Exercises

The diagram shows a rotation of x° about point R. A 13. Copy the diagram below. Then draw r(90°, P)(△ZXY).
rotation is rigid motion in which a figure and its image have Label the vertices of the image by using prime
the same orientation. notation.

R xЊ ZX
P
Example
Y
GHIJ has vertices G(0, −3), H(4, 1), I( −1, 2), and
14. What are the coordinates of r(180°, O)( - 4, 1)?
J( −5, −2). What arehtshme 1ve1rgtimcesseo_f0r(99c0°r,_Ot)1(G0H50IJ9)?.ai 15. WXYZ is a quadrilateral with vertices W(3, -1),

Use the rule r(90°, O)(x, y) = ( - y, x). X(5, 2), Y(0, 8), anhdsmZ(12,1-g1m).sGer_a0p9hcWr_XtY1Z0512.ai
r(90°, O)(G) = (3, 0)
r(90°, O)(H) = ( - 1, 4) and r(270°, O)(WXYZ).
r(90°, O)(I) = ( - 2, - 1)
r(90°, O)(J) = (2, - 5)

9-4  Compositions of Isometries

Quick Review Exercises

An isometry is a transformation that preserves distance. E 16. Sketch and describe the result of ഞ m
All of the rigid motions, translations, reflections, and reflecting E first across line / and
rotations, are isometries. A composition of isometries is then across line m.
also an isometry. All rigid motions can be expressed as a
composition of reflections. Each figure is an isometry image of the ng
figure at the right. Tell whether their
le
gn

le
The diagram shows a glide reflection N N an
of N. A glide reflection is an isometry in N a
which a figure and its image have a
opposite orientations. orientations are the same or opposite. hsm11gmse_09cr_t10524.ai

Then classify the isometry.

17. gle 18. ng 19. el hsm11gmse_09cr_t105

Example a

Describe the result of reflecting P first Phsm1C1g5m0Њsem_ഞ09cr_ 2t100. 5F△hi2nTs2dmA. aMt1hi1ehgiamms avsgeeert_oic0fe9RscyTh=r _(s-0mt2,1∘510)T1,5(A-g24(m6,40.,a)s1(ei△),_Ta0nA9dMcMhr)_.s(tm31,061)51.2g7m.asie_09cr_t10528.
across line Oand then across line m.

A composition of two reflections across
intersecting lines is a rotation. The angle

of rotation is twice the measure of the acute angle formed

by the intersecting lines. P is rotated 100° about C.

hsm11gmse_09cr_t10523.ai

604 Chapter 9  Chapter Review

9-5  Congruence Transformations

Quick Review Exercises Ly

Two figures are congruent if and only if there is a sequence 21. In the diagram at the right, MN x
of rigid motions that maps one figure onto the other. △LMN ≅ △XYZ. Identify a Z

Example congruence transformation that X
maps △LMN onto △XYZ. Y

Ry-axis(TGMB) = KWAV. y K 22. Fonts  Graphic designers p d
What are all of the congruent angles GW x use some fonts because they
and all of the congruent sides? have pleasing proportions or
T MA
A reflection is a congruence BV are easy to read from far away. The letters p and d to
transformation, so TGMB ≅ KWAV,
and corresponding angles and the right are used on a sign using a spgeeocmi1a2l_sfeo_nccts_.cA09rcer_t06.ai
corresponding sides are congruent. the letters congruent? If so, describe a congruence

∠T ≅ ∠K , ∠G ≅ ∠W , ∠M ≅ ∠A, and ∠B ≅ ∠V transformation that maps one ontogtehome1o2_tshe_ecrc.s_Icf09ncro_tt0,7.ai
TG = KW , GM = WA, MB = AV and TB = KV explain why not.

geom12_se_ccs_c09cr_t05.ai

9-6 Dilations

Quick Review Exercises

The diagram shows a 23. The blue figure is a dilation y

dilation with center C and image of the black figure. The 4
center of dilation is O. Tell Ϫ2 O
scale factor n. The preimage C a whether the dilation is an
and image are similar. na enlargement or a reduction.
Then find the scale factor.
In the coordinate plane, x
24
if the origin is the center of a dilation with scale factor n,

then P(x, y) S P′(nx, ny). hsm11gmse_09cr_t10510.ai Graph the polygon with the given vertices. Then graph
its image for a dilation with center (0, 0) and the given
Example
scale factor. hsm11gmse_09cr_t10521

The blue figure is a dilation image of the 4 24. M( -3, 4), A( -6, -1), T (0, 0), H(3, 2); scale factor 5
black figure. The center of dilation is A. Is
the dilation an enlargement or a reduction? 25. F( -4, 0), U(5, 0), N( -2, - 5); scale factor 1
2
What is the scale factor?
2 26. A dilation maps △LMN onto △L′M′N′. LM = 36 ft,
The image is smaller than the preimage, so LN = 26 ft, MN = 45 ft, and L′M′ = 9 ft. Find L′N′
and M′N′.
the dilation is a reduction. The scale factor is A
image length
original length = 2 2 4 = 62 , or 1 .
+ 3

hsm11gmse_09cr_t10517.ai

Chapter 9  Chapter Review 605

Lesson 9-7  Similarity Transformations

Quick Review Exercises

Two figures are similar if and only if there is a similarity 27. ▱GHJK has vertices G( -3, -1), H( -3, 2), J(4, 2),
transformation that maps one figure onto the other. and K(4, -1). Draw ▱GHJK and its image when you
apply the composition D2 ∘ Rx-axis.
When a figure is transformed by a composition of rigid
motions and dilations, the corresponding angles of the 28. Writing  Suppose that you have an 8 in. by 12 in.
image and preimage are congruent, and the ratios of photo of your friends and a 2 in. by 6 in. copy of the
corresponding sides are proportional. same picture. Are the two photos similar figures?
How do you know?
Example
29. Reasoning  A model airplane has an overall length
Is △JKL similar to △DCX ? If so, write a similarity
transformation rule. If not, explain why not. that is 1 the actual plane’s length, and an overall
20
J 1
height that is 18 the actual plane’s height. Are the
10
5 model a irplane and the actual airplane similar

figures? Explain.

30. Determine whether the figures below are similar. If

so, write the similarity transformation rule. If not,

K 7L explain. d

C p
5
X4

8
D

△JKL can be rotated and then translated so that J and D

coincide and JK and CD are collinear. Then if △JKL is

dilated by scale factor 45, then △JKL will coincide with geom12_se_ccs_c09cr_t09.ai
△DCX. So, the △JKL is similar to △DCX, and the similarity

transformation is a rotation, followed by a translation,
4
followed by a dilation of scale factor 5 .

606 Chapter 9  Chapter Review

9 Chapter Test athXM

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

FO

Do you know HOW? Determine whether the figures are similar. If so, identify
a similarity transformation that maps one to the other.
For Exercises 1–7, find the coordinates of the vertices of If not, explain.
the image of ABCD for each transformation.
18.  6 mm 6 mm

1. Rx = -4(ABCD) 1 y A x 4 mm 2 mm
2. T6-6, 87 (ABCD) O C D
3. r(90°, O)(ABCD)
Ϫ1
4. D 2 (ABCD)
3 Ϫ2 B 19.

5. (Rx = 0 ∘ T60, 57)(ABCD) Ϫ4

6. Ry = x(ABCD) geom12_se_ccs_c09ct_t01.ai

7. D3(ABCD) Identify the type of isometry that maps the black figure

hmsamps1P1(g-m4,s2e)_t0o9ct_t10570.atoi the blue figure.

20.
ut FWhat type of transformation has the same effect as

tueach composition of transformations?
8. Write the translation rule that
P′( -1, -1).
21. F

geom12_se_ccs_c09ct_t02.ai

9. Rx = 6 ∘ T60, -57 Do you UNDERSTAND?
10. T6-3, 27 ∘ T68, -47
11. Rx = 4 ∘ Rx = -2 22. Vhoscmab1u1lagrmy sIse_a0d9ilactti_otn1a0n5i7so9hm.asmei tr1y1? gExmpslaein_.09ct_t10580.ai
12. Ry = x ∘ Ry = -x
23. Writing  Line m intersects UH at N, and UN = NH.
What type(s) of symmetry does each figure have? Must H be the reflection image of U across line m?
Explain your reasoning.
13. 14. 15.
24. Coordinate Geometry  A dilation with center (0, 0)
and scale factor 2.5 maps (a, b) to (10,-25). What are
the values of a and b?

25. Error Analysis  A classmate says that a certain

regular polygon has 50° rotational symmetry.

Explain your classmate’s error.
ZvFeo(rr-tEi2chx,ese2smr)ocfif1os△r1esegXa1mYc6Zhsaewsni_idmt0h1i9l7vac,ehrtfri_isttnymitcd1ter10tsah15nXeg7s(cmf13oo,.raos4mire)d,_aYit0n(i9oa2tnc,e1.ths_)s,otma1fnt01hd51e7g2m.asie_09ct_ 2t61. 0mRq5ueu7aal4dstio.rpaanlinyinitn.gFg iCtnhdheotchooesoecrpodooinirndattisensAato,efBsA,oa,fBnAd, a′C,nBdin′,Ctahbneydf-iCrs2′t.by
16. (r(90°, O) ∘ Ry=1)(△XYZ) What composition of transformations maps △ABC

17. (r(180°, O) ∘ T62, -17)(△XYZ) onto △A′B′C′? Explain.

Chapter 9  Chapter Test 607

9 Common Core Cumulative ASSESSMENT
Standards Review

Some problems ask you to △G′H′K′ is the image of △GHK for a TIP 2
perform a transformation on a dilation with center (0, 0) and H′K′ = 8.
figure in the coordinate plane. What are the coordinates of H′? Identify the coordinates of
Read the sample question at G, H, and K from the graph.
the right. Then follow the tips to
answer it. K H2 y x Think It Through
Ϫ6 2
TIP 1 O The scale factor is
Ϫ2 H′K′
To calculate the scale HK = 8 = 2.
factor of a dilation, 4
you need to know the
lengths of a pair of G Ϫ4 To find the coordinates of H′,
corresponding sides. You
are given H′K′. You can ( -2, 4) (1, 2) multiply the coordinates of
use the graph to find HK. ( -1, -2) (4, 2)
# #H by the scale factor. H′ is at

(2 ( -1), 2 2), or ( -2, 4).
The correct answer is A.

hsm11gmse_09cu_t08727.ai
Selected Response
LVVeooscsacoabnubluarlayry Builder
Read each question. Then write the letter of the correct
As you solve problems, you must understand answer on your paper.

the meanings of mathematical terms. Match

each term with its mathematical meaning. 1. Which quadrilateral must have congruent diagonals?

A. centroid I. a transformation that kite parallelogram

B . circumcenter preserves distance rectangle rhombus

C . isometry II. the point of concurrency of 2. In a 30°-60°-90° triangle, the shortest leg measures
the medians in a triangle 13 in. What is the measure of the longer leg?

D . similarity III. an equation that states that 13 in. 1313 in.
transformation two ratios are equal

E. proportion IV. a mapping that may result 1312 in. 26 in.

F. transformation in a change in the position, 3. The vertices of ▱ABCD are A(1, 7), B(0, 0), C(7, -1),
and D(8, 6). What is the perimeter of ▱ABCD?
shape, or size of a figure

V. a composition of rigid 50 2200

motions and dilations 100 20 22

VI. the point of concurrency of
the perpendicular bisectors
of a triangle

608 Chapter 9  Common Core Cumulative Standards Review

4. Mica and Joy are standing at corner A of the 8. If you are given a line and a point not on the line, what
rectangular field shown below. is the first step to construct the line parallel to the
given line through the point?
A
Construct an angle from a point on the line to the
given point.

3 mi Draw a straight line through the given point.
4 Draw a ray from the given point that does not

intersect the line.

C 141 mi B Label a point on the given line, and draw a line
through that point and the given point

Mica walks diagonally across the field, from corner A 9. In a right triangle, which point lies on the hypotenuse?
to corner C. Joy walks from corner A to corner B, and
then to corner C. To the nearest hundredth of a mile, incenter centroid

hhoswmm1u1cghmfasreth_e0r9dcidu_Joty0w87al2k8th.aain Mica? orthocenter circumcenter

0.54 mi 1.46 mi 1 0. In △LMN, P is the centroid and LE = 24. What is PE?

1 mi 2 mi M

5. What is the area of the square floor tile? DE

L P N
5 V2 ft 8 F 10

50 ft2 100 12 ft2 9 16

100 ft2 150 ft2 11. Whhsamt i1s1thgemsusem_o0f9tchue_atn0g8le6m46ea.asiures of a 32-gon?

6. Whhsmat 1ty1pgemofssey_m0m9ectury_td0o8es645.ai 3200° 5400°

the figure have? 3800° 5580°

60° rotational symmetry 1 2. The diagonals of rectangle PQRS intersect at H. What is
90° rotational symmetry the length of QS?
line symmetry
point symmetry P 3x ϩ 5 Q
H
7. Which conditions allow you to conclude that a 4x Ϫ 1

quadrilateral is a parallelogram? thehostmhe1r1pgaimr osfe_09cu_t08735S.ai R
one pair of sides congruent, 6 23

sides parallel

perpendicular, congruent diagonals 12 46

diagonals that bisect each other hsm11gmse_09cu_t08647.ai

one diagonal bisects opposite angles

Chapter 9  Common Core Cumulative Standards Review 609

Constructed Response 19. Lucia makes a triangular garden in one corner of her
fenced rectangular backyard. She has 25 ft of edging
1 3. What is the value of x for which p } q? to use along the unfenced side of the garden. One of
p the fenced sides of the garden is 15 ft long. What is the
length, in feet, of the other fenced side of the garden?
115Њ
q 20. △DEB has vertices D(3, 7), E(1, 4), and B( -1, 5). In
which quadrant(s) is the image of r(270°, O)(△DEB)?
͑2x ϩ 5͒Њ Draw a diagram.

14. What is the measure of ∠H? 2 1. What is the area of an isosceles right triangle whose
hypotenuse is 712? Show your work.
AF
B 2 2. In △BGT, m∠B = 48, m∠G = 52, and GT = 6 mm.
hsm11gmse_09cu_t0865408Њ .aGi What is BT? Write your answer to the nearest

35Њ C H hundredth of a millimeter.

15. What is the area of the square, in square units, in the 23. In △ABC below, AB ≅ CB and BD # AC. Prove that
figure below? △ABD ≅ △CBD.

hsm11gymse_09cu_t08649.ai B

2 x
Ϫ4 Ϫ2 O 4

A DC

Ϫ3

16. In ▱PQRS, what is the value of x? Extended Response

Q R 24. Ihs s△mA1BC1gamrigshet_t0ri9acnugl_et?0J8u6st5if1y.yaoiur answer.

hsm11gmsexЊ_09cu_t08729.ai 4yA

84Њ
O

P 22Њ S Ϫ4 O x
4

1 7. For what value of x are the two triangles similar? B
C
5x Ϫ 2

2 h3sm11gmse_09cu_t10153.ai
4 25. LMNO has vertices L( -4, 0), M( -2, 3), N(1, 1), and
3x Ϫ 2 6
O( -1, -2). RSTV has vertices R(1, 1), S(3, -2),

18. Your friend is 5 ft 6 in. tall. When your friend’s shadow hTIssm(L6,M101N)g,Oamn≅dseVR_(S40T,93Vc)?u. GI_frtsa0op,8hw6tr5hit2ee.tatwhi eo quadrilaterals.
is 6 ft long, the shadow of a nearby sculpture is 30 ft rule for the

long.hWsmha1t i1sgthmeshee_ig0h9t,ciun_fet0et8, 6o5f t0h.eaisculpture? congruence transformation that maps LMNO to

RSTV. If not, explain why not.

610 Chapter 9  Common Core Cumulative Standards Review

Get Ready! CHAPTER

Skills Squaring Numbers and Finding Square Roots 10

Handbook, Simplify.
p. 888

1. 32 2. 82 3. 122 4. 152

5. 116 6. 164 7. 1100 8. 1169
Solve each quadratic equation.

9. x2 = 64 10. b2 - 225 = 0 11. a2 = 144

Review, Simplifying Radicals

p. 399 Simplify. Leave your answer in simplest radical form.

12. 18 13. 127 14. 175 15. 4 172

Lesson 1-8 Area

16. A garden that is 6 ft by 8 ft has a walkway that is 3 ft wide around it. What is the ratio
of the area of the garden to the area of the garden and walkway? Write your answer
in simplest form.

17. A rectangular rose garden is 8 m by 10 m. One bag of fertilizer can cover 16 m2. How
many bags of fertilizer will be needed to cover the entire garden?

Lessons 6-3 Classifying Quadrilaterals

and 6-5 Classify each quadrilateral as specifically as possible.

18. 19. 20.

Loohksmin1g1gAmshee_1a0dcoV_to09c8a9hb3smul1a1rgymse_10co_t098h9s4m11gmse_10co_t09895 611

21. A semiannual school fundraiser is an event that occurs every half year. What might
a semicircle look like in geometry?

22. A major skill is an important skill. How would you describe a major arc in
geometry?

23. Two buildings are adjacent if they are next to each other. What do you think
adjacent arcs on a geometric figure could be?

Chapter 10  Area

10CHAPTER Area

Download videos VIDEO Chapter Preview 1 Measurement
connecting math Essential Question  How do you
to your world.. 10-1 Areas of Parallelograms and Triangles find the area of a polygon or find the
10-2 Areas of Trapezoids, Rhombuses, circumference and area of a circle?
Interactive! ICYNAM
Vary numbers, ACT I V I TI and Kites 2 Similarity
graphs, and figures D 10-3 Areas of Regular Polygons Essential Question  How do perimeters
to explore math ES 10-4 Perimeters and Areas of Similar Figures and areas of similar polygons compare?
concepts.. 10-5 Trigonometry and Area
10-6 Circles and Arcs
10-7 Areas of Circles and Sectors
10-8 Geometric Probability

The online
Solve It will get
you in gear for
each lesson.

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Circles
Spanish English/Spanish Vocabulary Audio Online:
• Expressing Geometric Properties with
English Spanish Equations

adjacent arcs, p. 650 arcos adyacentes • Modeling with Geometry

Online access apothem, p. 629 apotema
to stepped-out
problems aligned arc length, p. 653 longitud de un arco
to Common Core
Get and view central angle, p. 649 ángulo central
your assignments
online. NLINE concentric circles, p. 651 círculos concéntricos
ME WO
O congruent arcs, p. 653 arcos congruentes
RK
HO diameter, p. 649 diámetro

major arc, p. 649 arco mayor

minor arc, p. 649 arco menor

Extra practice radius, pp. 629, 649 radio
and review
online sector of a circle, p. 661 sector de un círculo

segment of a circle, p. 662 segmento de un círculo

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Finding the Probability of Winning

The diagram below shows the target for a game at the school fair. The target is
composed of a regular octagon inscribed in a circle with center O. The length of
one side of the regular octagon is 9 in. Four vertices of the octagon are connected
to form a quadrilateral, as shown.
To play the game, a person throws one dart at the target. The player wins a prize if
the dart lands in any of the red or yellow regions.

9 in.

O

Task Description

Find the probability that a person who plays the game wins a prize. Round your
answer to the nearest whole percent. Assume the player’s dart is equally likely to
land at any point of the target.

Connecting the Task to the Math Practices MATHEMATICAL

As you complete the task, you’ll apply several Standards for Mathematical PRACTICES
Practice.

• You’ll use trigonometric ratios and a calculator to find lengths needed to
calculate areas. (MP 5)

• You’ll draw in auxiliary line segments to help you find areas. (MP 7)

• You’ll use more than one method to find areas and verify your results. (MP 1)

Chapter 10  Area 613

Concept Byte Transforming to MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Find Area
Use With Lessons Prepares for GM-AGFMS.D9.1A2..3G  -UGsMe vDo.l1u.m3 eUfosermvuolausme
10-1 and 10-2 formcyulliansdeforsr,cpyylirnadmeirdss,,pcyornaemsi,dasn, dcosnpehse,raensdtospshoelvres to
sporolvbelepmrosb. lems.
ACTIVITY
MP 7

You can use transformations to find formulas for the areas of polygons. In these
activities, you will cut polygons into pieces and use the pieces to form different
polygons.

1

Step 1 Count and record the number of units in the base and the height of the
parallelogram at the right.

Step 2 Copy the parallelogram onto grid paper.

Step 3 Cut out the parallelogram. Then cut it into two pieces as shown.

Step 4 Translate the triangle to the right through a distance equal to the
base of the parallelogram.

The translation results in a rectangle. Since their pieces are congruent, the
parallelogram and rectangle have the same area.

1. How many units are in the base of the rectangle? The height of the rectangle? hsm11gmse_1001a_t09484

2. How do the base and height of the rectangle compare to the base and height
of the parallelogram?

3. Write the formula for the area of the rectangle. Explain how you can use
this formula to find the area of a parallelogram.

2 hsm11gmse_1001a_t09486

Step 1 Count and record the number of units in the base and the height of the AB
triangle at the right.
AB
Step 2 Copy the triangle onto grid paper. Mark the midpoints A and B and hsm11gmse_1001a_t09490
draw midsegment AB.

Step 3 Cut out the triangle. Then cut it along AB.

Step 4 Rotate the small triangle 180° about the point B.
The bottom part of the triangle and the image of the top part form a
parallelogram.

4. How many units are in the base of the parallelogram? The height of
the parallelogram?

hsm11gmse_1001a_t09491

614 Concept Byte  Transforming to Find Area

5. How do the base and height of the parallelogram compare to the base and height
of the original triangle? Write an expression for the height of the parallelogram in
terms of the height h of the triangle.

6. Write your formula for the area of a parallelogram from Activity 1. Substitute the
expression you wrote for the height of the parallelogram into this formula. You
now have a formula for the area of a triangle.

3

Step 1 Count and record the bases and height of the trapezoid at the right. M N
Step 2 Copy the trapezoid. Mark the midpoints M and N, and draw

midsegment MN .

Step 3 Cut out the trapezoid. Then cut it along MN .
Step 4 Transform the trapezoid into a parallelogram.

7. What transformation did you apply to form a parallelogram?

8. What is an expression for the base of the parallelogram in terms of the two hsm11gmse_1001a_t09492
bases, b1 and b2, of the trapezoid?

9. If h represents the height of the trapezoid, what is an expression in terms of h
for the height of the parallelogram?

10. Substitute your expressions from Questions 8 and 9 into your area formula for a
parallelogram. What is the formula for the area of a trapezoid?

Exercises

11. In Activity 2, can a different rotation of the small triangle form a parallelogram?
If so, does using that rotation change your results? Explain.

12. Make another copy of the Activity 2 triangle. Find a rotation of the entire triangle
so that the preimage and image together form a parallelogram. How can you use
the parallelogram and your formula for the area of a parallelogram to find the
formula for the area of a triangle?

13. a. In the trapezoid at the right, a cut is shown from the midpoint N
of one leg to a vertex. What transformation can you apply to
the top piece to form a triangle from the trapezoid?

b. Use your formula for the area of a triangle to find a formula for
the area of a trapezoid.

14. Count and record the lengths of the diagonals, d1 and d2, of the d1
kite at the right. Copy and cut out the kite. Reflect half of the kite
across the line of symmetry d1 by folding the kite along d1. Use hsm11gmse_d12 001a_t09495
your formula for the area of a triangle to find a formula for the area
of a kite.

Concept Byte   Transforming to Find Area 615

10-1 Areas of Parallelograms MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
and Triangles
MG-AMFGS.9A1.12 .GU-sMe gGe.o1m.1e tUricseshgaepoemse, trhiecisrhmaepaesu, trhese,irand
tmheairspuroesp,eartnidesthtoeidr epsrcorpibeertoiebsjetoctds.escribe objects.
GM-AGFPSE..9B1.27. GU-sGePcEo.o2r.d7in  aUtsees ctoocrodminpautetes tpoerciommepteurtseof
peorlyimgoentesrsanodf paorelyagsoonfstrainadngalreesaasnodf rtericatnagnlgelsesa.nd
rMecPta3n,gMlesP. 4, MP 5, MP 6
MP 3, MP 4, MP 5, MP 6

Objective To find the area of parallelograms and triangles

A stage is being set up for a concert at the arena. 8 ft 8 ft
The stage is made up of blocks with tops that
You can combine are congruent right triangles. The tops of two of
triangles to make the blocks, when put together, make an 8 ft-by-
just about any 8 ft square. The band has requested that the stage
shape! be arranged to form the shape of an arrow. Draw a
diagram that shows how the stage could be laid out
MATHEMATICAL in the shape of an arrow with an area of at least
1000 ft2 but no more than 1400 ft2.
PRACTICES

Essential Understanding  You can find the area of a parallelogram or a triangle
when you know the length of its base and its height.

A parallelogram with the same base and height as a rectangle has the same
area as the rectangle.

Lesson Theorem 10-1  Area of a Rectangle

Vocabulary The area of a rectangle is the product of its base and height. h
• base of a A = bh

parallelogram b
• altitude of a
Theorem 10-2  Area of a Parallelogram
parallelogram
• height of a The area of a parallelogram is the product of a base and the

parallelogram corresponding height.
• base of a triangle A = bh
• height of a

triangle

hsm11gmh se_1001_t09119

b

A base of a parallelogram can be any one of its A lBhtaistsmued1e1gmse_1001_t09121
sides. The corresponding altitude is a segment
perpendicular to the line containing that base,
drawn from the side opposite the base. The
height is the length of an altitude.

616 Chapter 10  Area

hsm11gmse_1001_t09124

Problem 1 Finding the Area of a Parallelogram

Why aren’t the sides What is the area of each parallelogram?
ocof ntshiedepraerdalaleltliotgurdaems?
Altitudes must be A 4 in. B 3.5 cm
perpendicular to the 4.5 in. 4.6 cm
bases. Unless the
parallelogram is also a 5 in. 2 cm
rectangle, the sides are
not perpendicular to the You are given each height. Choose the corresponding side to use as the base.
bases.
A = bh Ah=s=mb2(h131.5g)m= s7e_1001_t09134

hs=m51(41)g=m2s0e _1001_Sutb0s9ti1tu3te2for b and h.

The area is 20 in.2. The area is 7 cm2.

Got It? 1. What is the area of a parallelogram with base length 12 m and height 9 m?

Problem 2 Finding a Missing Dimension

For ▱ABCD, what is DE to the nearest tenth? F 9 in.
DC
What does CF First, find the area of ▱ABCD. Then use the area formula a 13 in.
represent? second time to find DE. A EB
9.4 in.
CF is an altitude of the A = bh
parallelogram when = 13(9) = 117 Use base AD and height CF. hsm11gmse_1001_t09136

AD and BC are used as The area of ▱ABCD is 117 in.2.
bases.

A = bh

117 = 9.4(DE) Use base AB and height DE.

DE = 117 ≈ 12.4
9.4

DE is about 12.4 in.

Got It? 2. A parallelogram has sides 15 cm and 18 cm. The height corresponding to a
15-cm base is 9 cm. What is the height corresponding to an 18-cm base?

You can rotate a triangle about the midpoint of a side to form a parallelogram. 617

M
hh

bb
The area of the triangle is half the area of the parallelogram.

hsm11gmse_1001_t09137 Lesson 10-1 hAsmrea1s1ogfmPasrea_lle1l0o0gr1a_mt0s 9an1d38Triangles

Theorem 10-3  Area of a Triangle h
b
The area of a triangle is half the product of a base and the
corresponding height.

A = 12bh

A base of a triangle can be any of its sides. The corresponding height

is the length of the altitude to the line containing that base. hsm11gmse_1001_t09140

Problem 3 Finding the Area of a Triangle

Why do you need Sailing  You want to make a triangular sail like the one at 12 ft 2 in.
to convert the base the right. How many square feet of material do you need? 13 ft 4 in.
and the height into
inches? Step 1 Convert the dimensions of the sail to inches.
#(12 ft
You must convert #(13 ft 12 ifnt .) + 2 in. = 146 in. Use a conversion factor.
1
them both because
12 ifnt .) + 4 in. = 160 in.
you can only multiply 1

measurements with like Step 2 Find the area of the triangle.

units. A = 21bh

= 1 (160)(146) Substitute 160 for b and 146 for h.
2

= 11,680 Simplify.

# #Step 3 Convert 11,680 in.2 to square feet.
11,680 in.2 1 ft 1 ft = 8119 ft2
12 in. 12 in.
You need 8119 ft2 of material.
1 ft 1 in.
Got It? 3. What is the area of the triangle? 5 in. 1 ft

How do you know Problem 4 Finding the Area of ahnsmIrr1e1ggumlasre_F1ig0u0r1e_t09143

What is the area of the figure at the right?

the length of the base Find the area of each part of the figure. 8 in.
of the triangle? 6 in.
The lower part of the triangle area = 1 bh = 1 (6)8 = 24 in.2
figure is a square. The 2 2
base length of the square area = bh = 6(6) = 36 in.2

triangle is the same as area of the figure = 24 in.2 + 36 in.2 = 60 in.2
the base length of the

s quare. Got It? 4. Reasoning  Suppose the base lengths of the square and triangle in the

figure above are doubled to 12 in., but tahffeehcteeigdhh?tsomf e1a1cghmposley_go1n00re1m_at0in9s433.ai
the same. How is the area of the figure

618 Chapter 10  Area

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Find the area of each parallelogram.
PRACTICES

5. Vocabulary  Does an altitude of a triangle have to lie

1. 2. inside the triangle? Explain.
8 ft
10 m 8 ft 6. Writing  How can you show that a parallelogram and
20 m a rectangle with the same bases and heights have
equal areas?

Find the area of each triangle. 7. ▱ABCD is divided into two triangles along diagonal
AC. If you know the area of the parallelogram, how

3. hsm11g1m2 csme_1001_t0944. 34h.asim11gmse_1001_t09435.adi o you find the area of △ABC ?
D
8 in. C

16 cm

9 in.

AB

hsm11gmse_1001_t09436.ai
hsm11gmse_1001_t09437.ai

Practice and Problem-Solving Exercises PMRATAHCEMTAITCICEhASLsm11gmse_1001_t09438.ai

A Practice Find the area of each parallelogram. See Problem 1.
3.5 m
8. 9. 10.
15 cm 12 cm 5.8 m
4.7 in.
5.7 in.
20 cm 6 in.

4m

Find thhsemva1l1uge mofshef_o1r 0ea0c1h_pt0a9ra1ll4e8logram. See Problem 2.

11. h 12. hsm11gmse_100.501_t0915013. h13sm11ghmse_112001_t09152
14
0.3
8 10 h

0.4 18

Find the area of each triangle. See Problem 3.

14. 15. 4.5 hydsm11gm6syed_1001_t0911662. hsm11gmse_1001_t09165

5h.7smm 141mgmse5_m1001_t09156 3 ft

4m 3m 7.5 yd

2 ft 2 ft

hsm11gmse_1001_t09167 hsm11gmse_1001_t09170
hsm11gmse_1001_t09172

Lesson 10-1  Areas of Parallelograms and Triangles 619

17. Urban Design  A bakery has a 50 ft-by-31 ft parking lot. The four See Problem 4.
parking spaces are congruent parallelograms, the driving region is a
rectangle, and the two areas for flowers are 31 ft
congruent triangles. 15 ft

a. Find the area of the paved surface by adding 10 ft
the areas of the driving region and the four
parking spaces.

b. Describe another method for finding the
area of the paved surface.

c. Use your method from part (b) to find the
area. Then compare answers from parts (a)
and (b) to check your work.

B Apply 18. The area of a parallelogram is 24 in.2 and 50 ft
the height is 6 in. Find the length of the corresponding base.

19. What is the area of the figure at the right? 14 cm

64 cm2 88 cm2 96 cm2 112 cm2
8 cm
20. A right isosceles triangle has area 98 cm2. Find the length of each leg. HSM11GMSE_1001_a09795

21. Algebra  The area of a triangle is 108 in.2. A base and corresponding 1st pass 12-19-08 cm
heighDtuarkree 8

in the ratio 3 : 2. Find the length of the base and the corresponding height.

22. Think About a Plan  Ki used geometry

scoofntwstarurectteodc<rAeBa>taenthdeafpigouirnet at nttohoet<AorBing> ht<Aht.rBoS>.uhTgehhen Ck hsm11gmse_1001_t09175
C
D
she constructed line k parallel B

point C. Next, Ki constructed point D on line k

as well as AD and BD. She dragged point D

along line k to manipulate△ABD. How does

the area of △ABD change? Explain. A

• Which dimensions of the triangle change

when Ki drags point D?

• Do the lengths of AD and BD matter when calculating area?

23. Open-Ended  Using graph paper, draw an acute triangle, an obtuse triangle, and a
right triangle, each with area 12 units2. hsm11gmse_1001_t09427.ai

Find the area of each figure. 25. △BDJ y JK
24. ▱ABJF 27. ▱ BDKJ 4F
26. △DKJ 29. △BCJ
28. ▱ADKF 2
30. trapezoid ADJF
A BC D x

O 2 4 6 8 10 12

31. Reasoning  Suppose the height of a triangle is tripled. How does this affect the area

of the triangle? Explain. hsm11gmse_1001_t09177

620 Chapter 10  Area

For Exercises 32–35, (a) graph the lines and (b) find the area of the triangle
enclosed by the lines.

32. y = x, x = 0, y = 7 33. y = x + 2, y = 2, x = 6

34. y = -21 x + 3, y = 0, x = -2 35. y = 3 x - 2, y = - 2, x = 4
4

36. Probability  Your friend drew these three figures on a grid. A fly lands at random at
a point on the grid.

a. Writing  Is the fly more likely to land on one of the figures or on the blank

grid? Explain.

b. Slaunpdpoonsheosynmoeu1fik1gnugormewtsthheae_n1floy0n0laa1nn_dotst0ho9en4r?o2nE8ex.paolifatihne. figures. Is the fly more likely to

Coordinate Geometry  Find the area of a polygon with the given vertices.

37. A(3, 9), B(8, 9), C(2, -3), D( -3, -3) 38. E(1, 1), F(4, 5), G(11, 5), H(8, 1)

39. D(0, 0), E(2, 4), F(6, 4), G(6, 0) 40. K( -7, -2), L( -7, 6), M(1, 6), N(7, -2)

Find the area of each figure. 42. 21 cm 15 cm 43. 200 m
41. 25 ft 20 cm
120 m
25 ft 40 m

25 ft 60 m

C Challenge History  The Greek mathematician Heron is most famous for this formula for
the area of a triangle in terms of the lengths of its sides a, b, and c.

Ahs=m111s(gsm−sae)_(1s 0−01b)_(ts0−91c7),9wherhe ssm=1121 (gam+seb_+10c0)1_t09181.ai hsm11gmse_1001_t09183.ai

Use Heron’s Formula and a calculator to find the area of each triangle. Round
your answer to the nearest whole number.

44. a = 8 in., b = 9 in., c = 10 in. 45. a = 15 m, b = 17 m, c = 21 m

46. a. Use Heron’s Formula to find the area of this triangle. 1 15 in. 9 in.
2
b. Verify your answer to part (a) by using the formula A = bh.

12 in.

Lesson 10-1  Areas of Paralleloghrasmms1a1ngdmTrsiaen_g1le0s0 1_t091846.a2i 1

Standardized Test Prep

SAT/ACT 47. The lengths of the sides of a right triangle are 10 in., 24 in., and 26 in. What is the

area of the triangle?

116 in.2 120 in.2 130 in.2 156 in.2

Short 48. In quadrilateral ABCD, AB ≅ BC ≅ CD ≅ DA. Which type of quadrilateral could
ABCD never be classified as?
Response
square rectangle rhombus kite

49. Are the side lengths of △XYZ possible? Explain. X

6 4

Z

Y 11

PERFORMANCE TASK

Apply What You’ve Learned hsPMmRAT1AH1CEMgTAImTCICEsASeL _1001_t09429.ai

MP 5

Look back at the information given about the target on page 613. The diagram

of the target is shown again below, with three vertices of the regular octagon

labeled A, B, and C. BP is drawn perpendicular to AC.

AB

9 in. P
C

O


a. What is the measure of ∠ABC? Justify your answer.
b. Are the four red triangles congruent? Justify your answer.
c. What are the measures of the angles of △ABP?
d. Use a trigonometric ratio to find BP to the nearest hundredth of an inch.
e. Find AC to the nearest hundredth of an inch.
f. Use your results from parts (d) and (e) to find the area of △ABC. Round your

answer to the nearest tenth of a square inch.

622 Chapter 10  Area

10-2 Areas of Trapezoids, MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Rhombuses, and Kites GM-AMFGS.9A1.12 .GU-sMe gGe.o1m.1e tUricseshgaepoemse, trhiecisrhmaepaesu, trhese,irand
tmheairspuroesp,earntidesthtoeidr epsrcorpibeertoiebsjetoctds.escribe objects.

MP 1, MP 3, MP 4, MP 6

Objective To find the area of a trapezoid, rhombus, or kite

Rearranging Draw a trapezoid on a sheet of graph paper. b2
figures into
familiar shapes is Label the bases b1 and b2. Draw its midsegment. b1
an example of the Cut out the trapezoid, and then cut it along the
Solve a Simpler
Problem strategy. midsegment. Rotate the top part of the trapezoid
180° so that b1 and b2 now form one long base.
How can you use this new figure to find the area

of the trapezoid? Explain your reasoning.

MATHEMATICAL Essential Understanding  You can find the area of a trapezoid when you know
its height and the lengths of its bases.
PRACTICES

Lesson The height of a trapezoid is the perpendicular distance between the bases.
Theorem 10-4  Area of a Trapezoid
Vocabulary
• height of a

trapezoid

The area of a trapezoid is half the product of the height and b1
h
the sum of the bases.
b2
A = 1 h(b1 + b2)
2

Problem 1 Area of a Trapezoid

Which borders of Geography  What is the approximate area of Nevada? hsm11gmse_1002_t09235.ai
Nevada can you use
as the bases of a A = 1 h(b1 + b2) Use the formula for area of a 205 mi 309 mi
trapezoid? 2 trapezoid.

The two parallel sides of = 1 (309)(205 + 511) Substitute 309 for h, 205 for Reno 511 mi
2 b1, and 511 for b2.
Nevada form the bases Carson City

of a trapezoid. = 110,622 Simplify.

The area of Nevada is about 110,600 mi2. Las
Vegas

Got It? 1. What is the area of a trapezoid with height
7 cm and bases 12 cm and 15 cm?

Lesson 10-2  Areas of Trapezoidsh,sRm1h1ogmmsbe_u1s0e0s2,_aa0n9d88K4ites 623

Nevada

Problem 2 Finding Area Using a Right Triangle

What is the area of trapezoid PQRS? S 5m R

How are the sides You can draw an altitude that divides the trapezoid into a rectangle
related in a 30° -60°
-90° triangle? and a 30° -60° -90° triangle. Since the opposite sides of a rectangle are 60Њ
The length of the P 7m
hypotenuse is 2 times the congruent, the longer base of the trapezoid is divided into segments Q
length of the shorter leg,
#of lengths 2 m and 5 m. S 5m R
and the longer leg is
h = 2 13 longer leg = shorter leg 13
13 times the length of 1
the shorter leg. A = 2 h(b1 + b2) Use the trapezoid area formula.

= 1 (2 13 )(7 + 5) Substitute 2 13 for h, 7 for b1, and 5 for b2. hsm11hgmse_1002_t09237.ai
2
60Њ
= 1213 Simplify. P 2m 5m Q

The area of trapezoid PQRS is 1213 m2.

Got It? 2. Reasoning  In Problem 2, suppose h decreases so that m∠P = 45 while
angles R and Q and the bases stay the same. What is the area ohfsm11gmse_1002_t09241.ai
trapezoid PQRS?

Essential Understanding  You can find the area of a rhombus or a kite when
you know the lengths of its diagonals.

Theorem 10-5  Area of a Rhombus or a Kite

The area of a rhombus or a kite is half the product d2 d1

of the lengths of its diagonals. d1

A = 1 d1d2 d2
2

Rhombus   Kite

hsm 11Rghmosme_b1u00s2 _t09 24h3s .ami K1it1egmse_1002_t14116.ai

Problem 3 Finding the Area of a Kite

Do you need to know What is the area of kite KLMN? L
the side lengths of
the kite to find its Find the lengths of the two diagonals: 2m 3m
area? K 5m
KM = 2 + 5 = 7 m and LN = 3 + 3 = 6 m. M
No. You only need the 3m
A = 1 d1d2 Use the formula for area of a kite.
lengths of the diagonals. 2 Substitute 7 for d1 and 6 for d2.
1
= 2 (7)(6)

= 21 Simplify. N

The area of kite KLMN is 21 m2.

Got It? 3. What is the area of a kite with diagonals that ahrsem1121ignm. saen_d1090i2n_.t0lo9n24g4?.ai

624 Chapter 10  Area

Problem 4 Finding the Area of a Rhombus

How can you find the Car Pooling  The High Occupancy Vehicle (HOV) lane
length of AB? is marked by a series of “diamonds,” or rhombuses
painted on the pavement. What is the area of the HOV
AB is a leg of right lane diamond shown at the right?
△ABC. You can use the
Pythagorean Theorem, △ABC is a right triangle. Using the Pythagorean A
a2 + b2 = c2, to find 6.5 ft
its length. Theorem, AB = 26.52 - 2.52 = 6. Since the diagonals
B 2.5 ft C
of a rhombus bisect each other, the diagonals of the

HOV lane diamond are 5 ft and 12 ft.

A = 1 d1d2 Use the formula for area of a rhombus.
2
1
= 2 (5)(12) Substitute 5 for d1 and 12 for d2.

= 30 Simplify.

The area of the HOV lane diamond is 30 ft2.

Got It? 4. A rhombus has sides 10 cm long.
If the longer diagonal is 16 cm,
what is the area of the rhombus?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Find the area of each figure.
PRACTICES

7. Vocabulary  Can a trapezoid and a parallelogram

1. 4 m 2. 15 in. with the same base and height have the same area?

Explain.

6 m 18 in. hh

10 m 27 in. bb
4. 12 in.
3. 3 ft 8. Reasoning  Do you need to know all the side lengths
to find the area of a trapezoid?
hsm11gmse_1050f2t _t09504.ai hsm11gmse1_21i0n0. 2_t09511.ai
9.h Rsmea11sgomnisneg_1 0C0a2n_ty0o9u52f0in.adi the area of a rhombus if you
5. 10 m 6. 3 cm only know the lengths of its sides? Explain.

hsm11gmse_1002_t09512.ai 2 cm 2 cm 10. Reasoning  Do you need to know the lengths of the
sides to find the area of a kite? Explain.
10 m 20 m hsm11gmse_1002_t09514.ai
10 m 1 cm

hsm11gmse_1002_t09516.ai Lesson 10-2  Areas of Trapezoids, Rhombuses, and Kites 625

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Find the area of each trapezoid See Problem 1.
9 ft 6 ft
11. 21 in. 12. 24.3 cm 13.
18 ft
16 in. 8.5 cm
38 in. 9.7 cm

14. Find the area of a trapezoid with bases 12 cm and 18 cm and height 10 cm.
15. Fhisnmd1t1hgemasree_a10o0f2a_tt0ra9p24e6z.oaiid with basehssm2 1ft1agnmdse3_1ft0a0n2_dt0h9e2ig7h9.ta13i ft.
16. Geography  The border of Tennessee resembles a trapezoid with bases 3h4s0mm11igamnsde_1002_t09281.ai

440 mi and height 110 mi. Estimate the area of Tennessee by finding the area of the
trapezoid.

Find the area of each trapezoid. If your answer is not an integer, leave it in See Problem 2.
simplest radical form. 6m
8m
17. 5 ft 3 ft 18. 19.
6 ft 8 ft 10 m

60Њ

15 ft

Find the area of each kite. 21. 2hsmm11gmse_1002_t09284.ai See Problem 3.
3m 22. hsm11gmse_1002_t09283.ai
20. 2 in. 4m
hsm181ignm. se_1080i2n_. t09282.ai 3m 6 ft
8 in. 4 ft 4 ft

Find the area of each rhombus.20 ft 24. 10 in. See Problem 4.
23.
hsm181ignm. se_1002_t09288.ai 25.
hsm11gmse_1002_t09285.ai hsm11gmse_1002_t09290.ai
30 ft 6m

5m

B Apply 26. Think About a Plan  A trapezoid has two right angles, 12-m and 18-m bases, and an

8-m height. Sketch the trapezoid and find its perimeter and area.

••h smAH1roe1wgthmdesoere_isg1hk00nt 2ao_nwtg0inl9e2gs9tc1ho.aenihseeciguhtitvheeolphr soympo1up1ofgismnitdseet_ah1ne0g0pl2ee_srt?i0m92e9te3r.a?i hsm11gmse_1002_t09294.ai

626 Chapter 10  Area

27. Metallurgy  The end of a gold bar has the shape of a trapezoid 6.9 cm
with the measurements shown. Find the area of the end. 4.4 cm

28. Open-Ended  Draw a kite. Measure the lengths of its diagonals. 9.2 cm
Find its area.

Find the area of each trapezoid to the nearest tenth.

29. 30. 8 ft 9 ft 31.
3 cm 4 cm 30Њ 1.7 m 45Њ

2.1 m

3 cm 1 cm 0.9 m

Coordinate Geometry  Find the area of qhusma1d1rgilmatser_a1l00Q2R_St0T9.297.ai

32. y R 33. y R 34. y S
hsm11gmse_1002_t09296.ai hsm1R1gms2e_1002_t09298.ai
2 4
Q Ϫ2 O 2 x
Q 24 S x 2 Q Ϫ2 T

Ϫ2 T 2 S4 x

T

35. What is the area of the kite at the right? 9V2 m
hsm11gmse_1002_t4059Њ302.ai
90 m2 135 m2

h sm10181gmm2s e_1002_t09300.ai hsm11gmse_1002_ t2019630m1.2ai

36. a. Coordinate Geometry  Graph the lines x = 0, x = 6, y = 0, and y = x + 4. 6m
b. What type of quadrilateral do the lines form?
c. Find the area of the quadrilateral.

Find the area of each rhombus. Leave your answer in simplest radical form. hsm11gmse_1002_t09303.ai

37. 38. 39. 30Њ
45Њ 3 cm 60Њ 8 in.

4m

40. Visualization  The kite has diagonals d1 and 12dd2 1cdo2n. gruent to the sides of hthsme11gmse_1002_t09307.ai
rectangle. Explain why the area of the kite is
hsm11gmse_1002_t09305.ai d2
41. Draw a trapezoid. Label its bases b1 ahnsmd1b12gamnsde_i1ts00h2e_itg0h9t3h0.6T.ahi en draw a
diagonal of the trapezoid.

a. Write equations for the area of each of the two triangles formed. d1
b. Writing  Explain how you can justify the trapezoid area formula using

the areas of the two triangles.

Lesson 10-2  Areas of Trapezoids, Rhomhbsmus1e1sg,masned_1K0i0te2s_ t09308.6ai27

C Challenge 42. Algebra  One base of a trapezoid is twice the other. The height is the average of the
two bases. The area is 324 cm2. Find the height and the bases. (Hint: Let the smaller

base be x.) y

43. Sports  Ty wants to paint one side of 1
the skateboarding ramp he built. The y ϭ 0.25x2
ramp is 4 m wide. Its surface is
modeled by the equation y = 0.25x2. x

Use the trapezoids and triangles Ϫ2 Ϫ1 O 12 20 in.
shown to estimate the area to be 30Њ

painted. A 15 in. B
44. In trapezoid ABCD at the right, AB } DC. Find the area of ABCD.

hsm11gmse_1002_t09309.ai 135Њ C
D

SAT/ACT Standardized Test Prep

hsm11gmse_1002_t09310.ai
45. The area of a kite is 120 cm2. The length of one diagonal is 20 cm. What is the length

of the other diagonal?

12 cm 20 cm 24 cm 48 cm

46. △ABC ∼ △XYZ. AB = 6, BC = 3, and CA = 7. Which of the following are NOT
possible dimensions of △XYZ?

XY = 3, YZ = 1.5, ZX = 3.5 XY = 10, YZ = 7, ZX = 11

Short XY = 9, YZ = 4.5, ZX = 10.5 XY = 18, YZ = 9, ZX = 21
Response
47. Draw an angle. Construct a congruent angle and its bisector.

Mixed Review

48. Find the area of a right isosceles triangle that has one leg of length 12 cm. See Lesson 10-1.
49. A right isosceles triangle has area 112.5 ft2. Find the length of each leg. See Lesson 6-1.
50. Find the measure of an interior angle of a regular nonagon.
See Lesson 8-2.
G et Ready!  To prepare for Lesson 10-3, do Exercises 51–53. 10 m

Find the area of each regular polygon. Leave radicals in simplest form.

51. 52. 53.

10 cm 10 ft

hsm11gmse_1002_t09522.ai hsm11gmse_1002_t09523.ai

628 Chapter 10  Areahsm11gmse_1002_t09521.ai

10-3 Areas of Regular MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Polygons MG-AMFGS.9A1.12 .GU-sMe gGe.o1m.1e tUricseshgaepoemse, trhiecisrhmaepaesu, trhese,irand
tmheairspuroesp,earntidesthtoeidr epsrcorpibeertoiebsjetoctds.eAsclrsiobeGo-bCjeOc.tDs.13
Objective To find the area of a regular polygon MAlPso1M, MAPFS3.,9M12P.G4-,CMOP.46.1,3MP 7
MP 1, MP 3, MP 4, MP 6, MP 7

Solve a simpler You want to build a koi pond. For the border, you plan to use 3-ft-long
problem. Try using pieces of wood. You have 12 pieces that you can connect together at
fewer sides to see any angle, including a straight angle. If you want to maximize the area
what happens. of the pond, in what shape should you arrange the pieces? Explain your
reasoning.

MATHEMATICAL

PRACTICES
The Solve It involves the area of a polygon.

Lesson Essential Understanding  The area of a regular polygon Center
is related to the distance from the center to a side. Radius
Vocabulary Apothem
• radius of a You can circumscribe a circle about any regular polygon. The
center of a regular polygon is the center of the circumscribed
regular polygon circle. The radius of a regular polygon is the distance from the
• apothem center to a vertex. The apothem is the perpendicular distance
from the center to a side.

Problem 1 Finding Angle Measures hsm11gmse_1003_t09346.ai

The figure at the right is a regular pentagon with radii and an

How do you know apothem drawn. What is the measure of each numbered angle? 3
the radii make 21
isosceles triangles? m∠1 = 360 = 72 Divide 360 by the number of sides.
Since the pentagon is 5
a regular polygon, the
radii are congruent. So, m∠2 = 12m∠1 Tishoescaepleosthterimanbgilseefcotsrmtheed vertex angle of the
the triangle made by by the radii.
two adjacent radii and a    = 12(72) = 36
side of the polygon is an
isosceles triangle. 90 + 36 + m∠3 = 180 The sum of the measures of the angles of a triangle is 180.

     m∠3 = 54 hsm11gmse_1003_t09347
m∠1 = 72, m∠2 = 36, and m∠3 = 54.

Got It? 1. At the right, a portion of a regular octagon has radii 12
and an apothem drawn. What is the measure of each 3
numbered angle?

Lesson 10-3  Areas of Regular Polygons 629

Postulate 10-1
If two figures are congruent, then their areas are equal.

Suppose you have a regular n-gon with side s. The radii divide the figure a
into n congruent isosceles triangles. By Postulate 10-1, the areas of the s
isosceles triangles are equal. Each triangle has a height of a and a base
of length s, so the area of each triangle is 21as.

#Since there are n congruent triangles, the area of the n-gon is

A = n 21as. The perimeter p of the n-gon is the number of sides n times
the length of a side s, or ns. By substitution, the area can be expressed as
A = 12ap.

Theorem 10-6  Area of a Regular Polygon hsm11gmse_1003_t09349.ai

The area of a regular polygon is half the product of the apothem and a
the perimeter. p

A = 21ap

Problem 2 Finding the Area of a Regular Polygon hsm11gmse_1003_t09350.ai

What do you know What is the area of the regular decagon at the right? 12.3 in.
about the regular 8 in.
decagon? Step 1 Find the perimeter of the regular decagon. hsm11gmse_1003_t09351.ai

A decagon has 10 sides, p = ns Use the formula for the perimeter of an n-gon.
so n = 10. From the
diagram, you know = 10(8) Substitute 10 for n and 8 for s.

that the apothem a is = 80 in.

12.3 in., and the side Step 2 Find the area of the regular decagon.

length s is 8 in. A = 21ap

Use the formula for the area of a regular polygon.

= 12(12.3)(80) Substitute 12.3 for a and 80 for p.
= 492

The regular decagon has an area of 492 in.2.

Got It? 2. a. What is the area of a regular pentagon with an 8-cm apothem and
11.6-cm sides?

b. Reasoning  If the side of a regular polygon is reduced to half its length,

how does the perimeter of the polygon change? Explain.

630 Chapter 10  Area

Problem 3 Using Special Triangles to Find Area STEM

Zoology  A honeycomb is made up of regular hexagonal cells. The length of a side of
a cell is 3 mm. What is the area of a cell?

You know the length of a The apothem Draw a diagram to help find the
side, which you can use to apothem. Then use the area formula
find the perimeter. for a regular polygon.

Step 1 Find the apothem. 30Њ 60Њ
a 3 mm
The radii form six 60° angles at the center, so you can use a

#30°-60°-90° triangle to find the apothem.

a = 1.513 longer leg = 13 shorter leg

Step 2 Find the perimeter. 1.5 mm
p = ns Use the formula for the perimeter of an n-gon.
= 6(3) Substitute 6 for n and 3 for s.
= 18 mm

Step 3 Find the area. hsm11gmse_1003_t09353.ai

A = 21ap Use the formula for the area of a regular polygon.

  = 12(1.5 13) (18) Substitute 1.513 for a and 18 for p.

≈ 23.3826859 Use a calculator.

The area is about 23 mm2.

Got It? 3. The side of a regular hexagon is 16 ft. What is the area of the hexagon?
Round your answer to the nearest square foot.

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES
What is the area of each regular polygon? Round your
answer to the nearest tenth. 5. Vocabulary  What is the difference between a radius

and an apothem?

1. 2. 6. What is the relationship between the side length and

5 in. 3 ft the apothem in each figure?
a. a square
b. a regular hexagon
c. an equilateral triangle

3. 4. 7. Error Analysis  Your friend says you can use special

hsm11gms2em_1003_t09355.ai triangles to find the apothem of any regular polygon.

hsm11gmse_41V0033_t09357.ai What is your friend’s error? Explain.

Lesson 10-3  Areas of Regular Polygons 631
hsm11gmse_1003_t09360.aihsm11gmse_1003_t09362.ai

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Each regular polygon has radii and apothem as shown. Find the measure See Problem 1.
of each numbered angle.
7
8. 9. 10. 8

1 4 9
2
5
3 6

Find the area of each regular polygon with the given apothem a and side See Problem 2.

length s.

11. phesnmta1g1ognm, sae=_12040.33c_mt0,9s3=633.5a.i3 cm hsm11gmse_1120. 073-_gto0n9,3a6=4.a29i .1 ft, hs s=m2181fgtmse_1003_t09367.ai

13. octagon, a = 60.4 in., s = 50 in. 14. nonagon, a = 27.5 in., s = 20 in.

15. decagon, a = 19 m, s = 12.3 m 16. dodecagon, a = 26.1 cm, s = 14 cm

Find the area of each regular polygon. Round your answer to the nearest tenth. See Problem 3.

17. 18 ft 18. 19. 6m

8 in.

20. Art  You are painting a mural of colored equilateral triangles. The radius s
sohqfs uemaacr1eh1  gitnrmciahsne?g_le10is0132_.t70i9n3.6W8h.aait is thehasrmea1o1fgemacshe_tr1i0an0g3l_et0to93th6e9n.aeiarest 2

30Њ
hsm11gmse_1003_t09370.ai

Find the area of each regular polygon with the given radius or apothem. If your 12.7 in. s
answer is not an integer, leave it in simplest radical form.

21. 22.

8V3 in.
6 cm

23. 24. 25. hsm11gmse_1003_t09371.ai
4 in.
hsm11gmse_1003_t09373.ai
hs6mV131mgmse_1003_t09372.ai 5m

B Apply Find the measures of the angles formed by (a) two consecutive radii and

(b) a radius and a side of the given regular polygon.
hsm11gmse_1003_t09375.ai hsm11gmse_1003_t09376.ai
26. phesnmta1g1ognm se_1003_t092377. 4o.catai gon 28. nonagon 29. dodecagon

632 Chapter 10  Area

STEM 30. Satellites  One of the smallest space satellites ever developed has the shape of a
pyramid. Each of the four faces of the pyramid is an equilateral triangle with sides
about 13 cm long. What is the area of one equilateral triangular face of the satellite?
Round your answer to the nearest whole number.

31. Think About a Plan  The gazebo in the photo is built in the shape

of a regular octagon. Each side is 8 ft long, and the enclosed
area is 310.4 ft2. What is the length of the apothem?
• How can you draw a diagram to help you solve the problem?
• How can you use the area of a regular polygon formula?

32. A regular hexagon has perimeter 120 m. Find its area.

33. The area of a regular polygon is 36 in.2. Find the length of a side if

the polygon has the given number of sides. Round your answer to

the nearest tenth.

a. 3 b. 4 c. 6

d. Estimation  Suppose the polygon is a pentagon. What would

you expect the length of a side to be? Explain.

34. A portion of a regular decagon has radii and an apothem drawn. Find the 23
measure of each numbered angle. 1

35. Writing  Explain why the radius of a regular polygon is greater than the
apothem.

36. Constructions  Use a compass to construct a circle.

a. Construct two perpendicular diameters of the circle.

b. Construct diameters that bisect each of the four right angles. hsm11gmse_1003_t09377.a

c. Connect the consecutive points where the diameters intersect the circle. What

regular polygon have you constructed?

d. Reasoning  How can a circle help you construct a regular hexagon?

Find the perimeter and area of each regular polygon. Round to the nearest
tenth, as necessary.

37. a square with vertices at ( -1, 0), (2, 3), (5, 0) and (2, -3)
38. an equilateral triangle with two vertices at ( -4, 1) and (4, 7)
39. a hexagon with two adjacent vertices at ( -2, 1) and (1, 2)

40. To find the area of an equilateral triangle, you can use the s s
Vafaone.r rFmeifiqynuudltaihltaehAtfeeo=rararm12lebtaurhiloaaofnrAFgAilge=u=i41srse21t2ao11puu.3ssAeiinntthghtiwetrhdfoeowwrfmoaaryyumstloauasflAianfo=Adll41to=hsw2e12s1ba:hr3e..a of
b. Find the area of Figure 2 using the formula A = 12ap. s s
2 2
41. For Problem 1 on page 629, write a proof that the apothem
Proof bisects the vertex angle of an isosceles triangle formed by two radii. Figure 1 Figure 2

hsm11gmse_1003_t09381.ai 633
Lesson 10-3  Areas of Regular Polygons

C Challenge 42. Prove that the bisectors of the angles of a regular polygon are concurrent and that
Proof they are, in fact, radii of the polygon. (Hint: For regular n-gon ABCDE . . ., let P be

the intersection of the bisectors of ∠ABC and ∠BCD. Show that DP must be the
bisector of ∠CDE.)

43. Coordinate Geometry  A regular octagon with center at the y

origin and radius 4 is graphed in the coordinate plane. V2

a. Since V2 lies on the line y = x, its x- and y-coordinates are 2 V1 (4, 0)
equal. Use the Distance Formula to find the coordinates of V2 2x
to the nearest tenth. Ϫ2 O
12bh Ϫ2
b. Use the coordinates of V2 and the formula A = to find
the area of △V1OV2 to the nearest tenth.
c. Use your answer to part (b) to find the area of the octagon to

the nearest whole number.

Standardized Test Prep

SAT/ACT 44. What is the area of a regular pentagon with an apothem of 25.1 mm ahnsdmp1e1rgimmestee_r1003_t09382.ai

of 182 mm?

913.6 mm2 2284.1 mm2 3654.6 mm2 4568.2 mm2

45. What is the most precise name for a regular polygon with four right angles?

square parallelogram trapezoid rectangle
Short
46. △ABC has coordinates A( -2, 4), B(3, 1), and C(0, -2). If you reflect △ABC across
Response the x-axis, what are the coordinates of the vertices of the image △A′B′C′?

A′(2, 4), B′( -3, 1), C′(0, -2) A′(4, -2), B′(1, 3), C′( -2, 0)

A′( -2, -4), B′(3, -1), C′(0, 2) A′(4, 2), B′(1, -3), C′( -2, 0)

47. An equilateral triangle on a coordinate grid has vertices at (0, 0) and (4, 0). What are
the possible locations of the third vertex?

Mixed Review

48. What is the area of a kite with diagonals 8 m and 11.5 m? See Lesson 10-2.

49. The area of a trapezoid is 42 m2. The trapezoid has a height of 7 m and one base of 4 m.
What is the length of the other base?

G et Ready!  To prepare for Lesson 10-4, do Exercises 50–52. See Lesson 1-8.
6 cm
Find the perimeter and area of each figure.

50. 51. 52.

4m

7 in. 8 m 8 cm

634 Chapter 10  Area hsm11gmse_1003_t09385.ai hsm11gmse_1003_t09387.ai
hsm11gmse_1003_t09383.ai

10-4 Perimeters and Areas CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

of Similar Figures Prepares for GM-AGFMS.D9.1A2..3G  -UGsMe vDo.l1u.m3 eUfosermvuolausme
formcyulliansdeforsr,cpyylirnadmeirdss,,pcyornaemsi,dasn, dcosnpehse,raensdtospshoelvres to
sporolvbelepmrosb. lems.

MP 1, MP 3, MP 4, MP 5, MP 7, MP 8

Objective To find the perimeters and areas of similar polygons

On a piece of grid paper, draw a 3 unit-by-4 unit rectangle. Then draw
three different rectangles, each similar to the original rectangle. Label
them I, II, and III. Use your drawings to complete a chart like this.

You already Rectangle Perimeter Area
know that if you
double the length Original ■ ■
and width of a I ■ ■
rectangle, its area II ■ ■
quadruples. III ■ ■

MATHEMATICAL Use the information from the first chart to complete a chart like this.

PRACTICES

Scale Ratio of Ratio of
Areas
Rectangle Factor Perimeters
■
I to Original ■ ■ ■
■
II to Original ■ ■

III to hOsrmigi1n1agl mse_1■004_a09810■.ai

How do the ratios of perimeters and the ratios of areas compare with
the scale factors?

In the Solve It, you compared the areas of simhislamr f1ig1ugrems.se_1004_a09887.ai

Essential Understanding  You can use ratios to compare the perimeters and
areas of similar figures.

Theorem 10-7  Perimeters and Areas of Similar Figures

If the scale factor of two similar figures is a , then
b
a
(1) the ratio of their perimeters is b and

(2) the ratio of their areas is a2 .
b2

Lesson 10-4  Perimeters and Areas of Similar Figures 635

Problem 1 Finding Ratios in Similar Figures

The trapezoids at the right are similar. The ratio of the lengths of 6m
9m
How do you find the corresponding sides is 96, or 2 .
s cale factor? 3
Write the ratio of
the lengths of two A What is the ratio (smaller to larger) of the perimeters?
corresponding sides.
The ratio of the perimeters is the same as the ratio of corresponding
2
sides, which is 3 .

B What is the ratio (smaller to larger) of the areas?

The ratio of the areas is the square of the ratio of corresponding sides,
22 4
which is 32 , or 9 .

Got It? 1. Two similar polygons have corresponding sides in the ratio 5 : 7.

a. What is the ratio (larger to smaller) of their perimeters? hsm11gmse_1004_t09322.ai
b. What is the ratio (larger to smaller) of their areas?

When you know the area of one of two similar polygons, you can use a proportion to
find the area of the other polygon.

Problem 2 Finding Areas Using Similar Figures

Can you eliminate Multiple Choice  The area of the smaller regular pentagon is about
a ny answer choices 27.5 cm2. What is the best approximation for the area of the larger
immediately? regular pentagon?
Yes. Since the area of
the smaller pentagon 11 cm2 69 cm2 172 cm2 275 cm2 4 cm 10 cm
is 27.5 cm2, you know
that the area of the Regular pentagons are similar because all angles measure 108 and all side lengths is 140,
larger pentagon must be hsm11gmse_1004_t09323.ai
greater than that, so you sides in each pentagon are congruent. Here the ratio of corresponding
can eliminate choice A. 2 22 4 areas.
or 5 . The ratio of the areas is 52 , or 25 .

4 = 27.5 Write a proportion using the ratio of the
25 A

4A = 687.5 Cross Products Property

A = 687.5 Divide each side by 4.
4

A = 171.875 Simplify.

The area of the larger pentagon is about 172 cm2. The correct answer is C.

Got It? 2. Tphareasllcealloegfraacmtoirso9f6twino.2s.iWmhilaatr ipsatrhaellaerloeagroafmthseissm34. aTlhleerapraeraaollfetlhoegrlaamrg?er

636 Chapter 10  Area