Geometry

In Lesson 6-8, you wrote a plan for the proof of the Trapezoid Midsegment Theorem.
Now you will write the full coordinate proof.

Proof Problem 2 Writing a Coordinate Proof

Refer to the plan from Write a coordinate proof of the Trapezoid Midsegment Theorem. y
Lesson 6-8. Find the R(2b, 2c) A(2d, 2c)
coordinates of M and N. Given: MN is the midsegment of trapezoid ORAP.
Determine whether MN
is parallel to OP and RA. Prove: MN } OP , MN } RA , MN = 1 (OP + RA) M N
Then find and compare 2 O x
the lengths of MN, OP,
and RA. Coordinate Proof: P(2a, 0)

Use the Midpoint Formula to find the coordinates of M and N.

M=( ) 2b + 0, 2c + 0 = (b, c)
2 2

N=( ) 2a + 2d, 0 + 2c = (a + d, c) hsm11gmse_0609_t06330
2 2

Use the Slope Formula to determine whether MN is parallel to OP and RA.
slope of MN = (a +c -d)c- b = 0
slope of RA = 22dc -- 22cb = 0
slope of OP = 20a--00 = 0

The three slopes are equal, so MN } OP and MN } RA.

Use the Distance Formula to find and compare MN, OP, and RA.

MN = 2[(a + d) - b)]2 + (c - c)2 = a + d - b
OP = 2(2a - 0)2 + (0 - 0)2 = 2a
RA = 2(2d - 2b)2 + (2c - 2c)2 = 2d - 2b

MN ≟ 1 (OP + RA) Check that MN = 1 (OP + RA) is true.
2 2

a + d - b ≟ 1 [2a + (2d - 2b)] Substitute.
2

a + d - b = a + d - b  ✔ Simplify.

So,  (1)  the midsegment of a trapezoid is parallel to its bases, and
(2)  the length of the midsegment of a trapezoid is half the sum of the

lengths of the bases.

Got It? 2. Write a coordinate proof of the Triangle Midsegment Theorem
(Theorem 5-1).

Lesson 6-9  Proofs Using Coordinate Geometry 415

Lesson Check Do you UNDERSTAND? MATHEMATICAL

Do you know HOW? PRACTICES

1. Use coordinate geometry to prove that the diagonals 2. Reasoning  Describe a good strategy for placing the
of a rectangle are congruent.
vertices of a rhombus for a coordinate proof.
a. Place rectangle PQRS in the coordinate plane with
P at (0, 0). 3. Error Analysis  Your y
classmate places P(b, c) Q(a − b, c)
b. What are the coordinates of Q, R, and S? a trapezoid on the
c. Write the Given and Prove statements. coordinate plane. x
d. Write a coordinate proof. What is the error? O R(a, 0)

Practice and Problem-Solving Exercises MATHEMATICAL hsm11gmse_0609_t06602

PRACTICES See Problems 1 and 2.

A Practice Developing Proof  Complete the following coordinate proofs.

4. The diagonals of an isosceles trapezoid are congruent. F(؊b, c) y G(b, c)

Given:  Trapezoid EFGH with EF ≅ GH E(؊a, 0) O x
H(a, 0)
Prove:  EG ≅ FH
a. Find EG.
b. Find FH.
c. Explain why EG ≅ FH.

5. The medians drawn to the congruent sides of an isosceles triangle are congruent.

Given:  △PQR with PQ ≅ RQ, M is the midpoint of hsm11gmse_060y9_Qt0(06, 62b0)3
PQ, N is the midpoint of RQ

Prove:  PN ≅ RM M N
a. What are the coordinates of M and N?
x
b. What are PN and RM? R(2a, 0)

c. Explain why PN ≅ RM. P(؊2a, 0) O

B Apply Tell whether you can reach each type of conclusion below using coordinate

methods. Give a reason for each answer.

6. AB ≅ CD 7. AB } CD 8. AB # CD
9. AB bisects CD. 10. AB bisects ∠CAD.
hs1m1.1 ∠1gAm≅s∠e_B0609_t06604

12. ∠A is a right angle. 13. AB + BC = AC 14. △ABC is isosceles.

15. Quadrilateral ABCD is a rhombus. 16. AB and CD bisect each other.

17. ∠A is the supplement of ∠B. 18. AB, CD, and EF are concurrent.

416 Chapter 6  Polygons and Quadrilaterals

19. Flag Design  The flag design at the right is made by connecting
Proof the midpoints of the sides of a rectangle. Use coordinate
geometry to prove that the quadrilateral formed is a rhombus.

20. Open-Ended  Give an example of a statement that you think
is easier to prove with a coordinate geometry proof than
with a proof method that does not require coordinate
geometry. Explain your choice.

Use coordinate geometry to prove each statement.

Proof

21. Think About a Plan  If a parallelogram is a rhombus, its
diagonals are perpendicular (Theorem 6-13).

• How will you place the rhombus in a coordinate plane?
• What formulas will you need to use?

22. The altitude to the base of an isosceles triangle bisects the base.

23. If the midpoints of a trapezoid are joined to form a quadrilateral, then the
quadrilateral is a parallelogram.

24. One diagonal of a kite divides the kite into two congruent triangles.

25. You learned in Theorem 5-8 that the centroid of a triangle is two thirds the distance
Proof from each vertex to the midpoint of the opposite side. Complete the steps to prove

this theorem.

a. Find the coordinates of points L, M, and N, the midpoints of the y B(6q, 6r)
L PM
b. Fsiidnedseoqfu△aAtioBnCs. of <AM>, <BN>, and <CL>. of <AM> and <BN>.
c. Find the coordinates of <pCoL>i.nt P, the intersection
d. Show that point P is on

e. Use the Distance Formula to show that point P is two thirds the O x
distance from each vertex to the midpoint of the opposite side. A N C(6p, 0)

26. Complete the steps to prove Theorem 5-9. You are given △ABC y
Proof with altitudes p, q, and r. Show that p, q, and r intersect at a point
C(0, c) p
(called the orthocenter of the triangle). r
ba.. SThheowslothpaetothf eBCeqiusa-tcibo.nWohfalitnise
the slope of line p? hsm11gmse_0609_t06605
b (x - a).
p is y = c

c. What is the equation of line q? A(a, 0) B(b, 0)
( ) d. Show that lines p and q intersect at 0, -cab . x
Oq
e. The slope of AC is -ca. What is the slope of line r?
f. Show that the equation of line r is a
y = c (x - b).
( ) g. Show that lines r and q intersect at 0, -cab .
h. What are the coordinates of the orthocenter of △ABC? hsm11gmse_060ᐉ9_t06606

C Challenge 27. Multiple Representations  Use the diagram at the right.11 c a
a. Explain using area why 2 ad = 2 bc and therefore ad = bc. d p

b. Find two ratios for the slope of /. Use these two ratios to show b
that ad = bc.

Lesson 6-9  Proofs Using Coordinate Geometry 417

hsm11gmse_0609_t06608

28. Prove: If two lines are perpendicular, the product of their slopes is -1. ᐉ2 y
Proof a. Two nonvertical lines, /1 and /2, intersect as shown at the right. B(■, ■)

Find the coordinates of C. 3 A(a, b) ᐉ1
1 x
b. Choose coordinates for D and B. (Hint: Find the relationship D(■, ■) 2
between ∠1, ∠2, and ∠3. Then use congruent triangles.) O
C(■, ■)
c. Complete the proof that the product of slopes is -1.

PERFORMANCE TASK

Apply What You’ve Learned hsm11PMgRAmTAHCEsMeTAI_TCI0CEA6SL09_t06607

MP 3

Look back at the information about Alejandro’s kite on page 351. In the Apply

What You’ve Learned in Lesson 6-6, you showed that the shape of Alejandro’s

kite fits the geometric definition of a kite.

You can use coordinate geometry to find a formula for the area of a kite. Use
the diagram below, in which PRSO is a kite with one vertex at the origin and
diagonal OR on the x-axis. In the diagram, OP = OS and PR = SR.

y

P

x
OR

S



a. If OR = a, what are the coordinates of R?
b. If the coordinates of P are (b, c), what are the coordinates of S? Explain how

you know.

c. Explain why the area of PRSO = ac.
d. a is the length of one diagonal of the kite PRSO. What is c with respect to the kite?

e. In words, an area formula for a triangle is “half the product of the base and height.”
How can you express an area formula for a kite in words?

418 Chapter 6  Polygons and Quadrilaterals

6 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 6-6
problems you and 6-9. Use the work you did to complete the following.
will pull together
many concepts 1. Solve the problem in the Task Description on page 351 by finding the length of the
and skills that vertical support for the kite and the area of the paper used to make the kite. Show all
you have learned your work and explain each step of your solution.
about polygons.
2. Reflect  Choose one of the Mathematical Practices below and explain how you
applied it in your work on the Performance Task.
MP 3: Construct viable arguments and critique the reasoning of others.

MP 4: Model with mathematics.

On Your Own

Alejandro’s friend Amy also draws a sketch for a paper kite. In her sketch shown below,
she colored the green region so that the horizontal support MO bisects two angles of the
quadrilateral MNOQ. Like Alejandro, Amy forgets to show the vertical support for the kite.

N

15 in. 15 in.

M 24 in. O

20 in. Q 20 in.
P

Find the length of the missing vertical support for Amy’s kite and the areas of the green and
purple papers used in the kite.

Chapter 6  Pull It All Together 419

6 Chapter Review

Connecting and Answering the Essential Questions

1 Measurement Polygon Angle-Sum Theorems
You can find the sum (Lesson 6-1)
of the interior angle
measures of any polygon Sum = (n - 2)180, where n is the
using a formula based on number of sides
its number of sides.
Parallelograms Special Parallelograms
2 Reasoning (Lessons 6-2 and 6-3) (Lessons 6-4 and 6-5)
and Proof
If you know certain  
information about
the sides, angles,   Rhombus  Rectangle  Square
or diagonals of a
quadrilateral, you can Trapezoids and Kites (Lesson 6-6)
classify it.
hsm 11gmshe_sm061c1rg_tm06she3s_4m0661c1rg_mt0s6e3_4076crh_Cts0om6o1r3d14ig8nmatesePh_rs0omo6fc1sr1_(Lgte0ms6ss3oen4h_9s60m-691)cr1_gtm06s3e5_006cr_t06
3 Coordinate
Geometry y y
Coordinate proofs use
variable coordinates to   Trapezoid Kite B(0, b) C(a, b) C(0, b)
prove relationships in the
coordinate plane. Apphly(siLnmegs1sC1oongosmr6ds-i7nea_aht0nesd6mGc6er1-_o81tm)g0e6mt3rsy5e2_06cr_t0635O3 x Ox
( )midpoint: A(a, 0) B(؊a, 0) A(a, 0)

x1 + x2, y1 + y2
2 2

distance: 5slo(pxe2:-xy22x--1)2xy11+ (y2 - y1)2 hsm11gmse_06crh_stm061315g5mse_06cr_t06356

Chapter Vocabulary

• base, base angle, and leg • equiangular, equilateral • midsegment of a • rectangle (p. 375)
of a trapezoid (p. 389) polygon (p. 354) trapezoid (p. 391) • regular polygon (p. 354)
• rhombus (p. 375)
• consecutive angles • isosceles trapezoid • opposite angles (p. 359) • square (p. 375)
(p. 360) (p. 389) • opposite sides (p. 359) • trapezoid (p. 389)
• parallelogram (p. 359)
• coordinate proof (p. 408) • kite (p. 392)

Choose the vocabulary term that correctly completes the sentence.
1. A parallelogram with four congruent sides is a(n) ? .
2. A polygon with all angles congruent is a(n) ? .
3. Angles of a polygon that share a side are ? .
4. A(n) ? is a quadrilateral with exactly one pair of parallel sides.

420 Chapter 6  Chapter Review

6-1  The Polygon Angle-Sum Theorems

Quick Review Exercises

The sum of the measures of the interior angles of an n-gon Find the measure of an interior angle and an exterior
angle of each regular polygon.
is (n - 2)180. The measure of one interior angle of a regular
-
n-gon is (n 2)180 . The sum of the measures of the exterior 5. hexagon 6. 16-gon 7. pentagon
n

angles of a polygon, one at each vertex, is 360.

Example 8. What is the sum of the exterior angles for each
polygon in Exercises 5–7?
Find the measure of an interior angle of a regular 20‑gon.
Find the measure of the missing angle.

Measure = (n - 2) 180 Corollary to the Polygon Angle-Sum 9. xЊ 83Њ 10. 122Њ
n Theorem
89Њ

# = (20 -202)180 Substitute. 119Њ zЊ
79Њ

= 18 180 Simplify.
20

= 162

The measure of an interior angle is 162. hsm11gmse_06cr_t06359
hsm11gmse_06cr_t06357

6-2  Properties of Parallelograms

Quick Review Exercises

Opposite sides and opposite angles of a parallelogram Find the measures of the numbered angles for each
are congruent. Consecutive angles in a parallelogram are parallelogram.
supplementary. The diagonals of a parallelogram bisect
each other. If three (or more) parallel lines cut off congruent 11. 3 38Њ 2 12. 1 2
segments on one transversal, then they cut off congruent 1 99Њ 79Њ 3
segments on every transversal.
13. hsm113 gm1s63eЊ_06cr_t061346. 11 2 3
Example
37Њ 2 hsm11gmse_06cr_t06363
Find the measures of the numbered angles in
the parallelogram. Find the values of x and y in ▱ABCD.

23 15. AAhBBsm==1221yy,g+BmC1,s=BeC_y0+=63yc,r+C_Dt10, =C6D356x=5h-7s1xm,-D1A31,g=DmA2xs=e+_34x06cr_t06366
1 56Њ 16.

Since consecutive angles are supplementary,
m∠1 = 180 - 56, or 124. Since opposite angles are
congruent, m∠2 = 56 and m∠3 = 124.

hsm11gmse_06cr_t06360

Chapter 6  Chapter Review 421

6-3  Proving That a Quadrilateral Is a Parallelogram

Quick Review Exercises

A quadrilateral is a parallelogram if any one of the following Determine whether the quadrilateral must be a
is true. parallelogram.

• Both pairs of opposite sides are parallel. 17. 18.
• Both pairs of opposite sides are congruent.
• Consecutive angles are supplementary. Algebra  Find the values of the variables for which ABCD
• Both pairs of opposite angles are congruent.
• The diagonals bisect each other. must be a parallelogram.
• One pair of opposite sides is both congruent
19. Bhsm(3y1Ϫ1g20m)Њse_C06cr_t062307. 0hBsm141x gϪm2 se3x_06Ccr_t06372
and parallel. (4y ϩ 4)Њ 3y Ϫ 1
3y Ϫ 3
Example 4xЊ (2x ϩ 6)Њ
A D
Must the quadrilateral be a parallelogram? AD

Yes, both pairs of opposite angles are
congruent.

6-4  Properties of Rhombusesh,sRme11cgtamnseg_l0e6sc,r_ath0ns6m3d618S1gqmusae_r0e6scr_t06373hsm11gmse_06cr_t06375

Quick Review Exercises

A rhombus is a parallelogram with four congruent sides. Find the measures of the numbered angles in each special
parallelogram.
A rectangle is a parallelogram with four right angles.

A square is a parallelogram with four congruent sides and 21. 12 22. 1 2 3
four right angles. 3 56Њ

The diagonals of a rhombus are perpendicular. Each 32Њ
diagonal bisects a pair of opposite angles.

The diagonals of a rectangle are congruent.

Determine whether each statement is always, sometimes,

Example 2 or never true. hsm11gmse_06cr_t06381.ai
13
What are the measures of the numbered 23. hAsrmho1m1bgums issea_s0q6ucarr_et. 06380.ai
angles in the rhombus? 60Њ
24. A square is a rectangle.

m∠1 = 60 Each diagonal of a rhombus 25. A rhombus is a rectangle.
bisects a pair of opposite angles.
m∠2 = 90 The diagonals of a rhombus are #. 26. The diagonals of a parallelogram are perpendicular.

60 + m∠2 + m∠3 = 180 Triangle Angle-SumhsThmm1. 1gmse_06cr_ 2t07.6 T3h7e9d.aiai gonals of a parallelogram are congruent.
60 + 90 + m∠3 = 180 Substitute. 28. Opposite angles of a parallelogram are congruent.

m∠3 = 30 Simplify.

422 Chapter 6  Chapter Review

6-5  Conditions for Rhombuses, Rectangles, and Squares

Quick Review Exercises

If one diagonal of a parallelogram bisects two angles of Can you conclude that the parallelogram is a rhombus,
the parallelogram, then the parallelogram is a rhombus. rectangle, or square? Explain.
If the diagonals of a parallelogram are perpendicular,
then the parallelogram is a rhombus. If the diagonals of a 29. 30.
parallelogram are congruent, then the parallelogram is a
rectangle. For what value of x is the figure the given parallelogram?
Justify your answer.
Example
31. h smRh1o1mgbums se_06cr_t063328. 3h.asim11Regcmtansgel_e06cr_t06384.ai
Can you conclude that the parallelogram is a rhombus,
rectangle, or square? Explain. 22x Ϫ 1 xϩ3

Yes, the diagonals are perpendicular, 2
so the parallelogram is a rhombus.

(5x Ϫ 30)Њ (3x ϩ 6)Њ

6-6  Trapezoids and Kites hsm11gmse_06cr_t06h3s8m21.a1igmse_06cr_t06386.ahism11gmse_06cr_t06385.ai

Quick Review Exercises

The parallel sides of a trapezoid are its bases and the Find the measures of the numbered angles in each
nonparallel sides are its legs. Two angles that share a isosceles trapezoid.
base of a trapezoid are base angles of the trapezoid.
The midsegment of a trapezoid joins the midpoints 33. 12 34. 1
of its legs. 80Њ
45Њ 3 2
The base angles of an isosceles trapezoid are congruent. The 3
diagonals of an isosceles trapezoid are congruent.

The diagonals of a kite are perpendicular. Find the measures of the numbered angles in each kite.

Example 35. h sm11gmse_06cr_t063368. 83.a4Њi 1 38Њ

ABCD is an isosceles trapezoid. BC 1
What is m∠C?
2 hsm11gm2 se_06cr_t06389.ai

65Њ

Since BC } AD, ∠C and ∠D are A 60Њ 37. Algebra  A trapezoid has base lengths of
same-side interior angles. D
l(e6nxg-th1o)fu(n5ixts-a3n)du3nuitnsi.tWs. hItastmhissidtmhsee1gv1magleumnetsohefax_s?0a6cr_t06391.ai
m ∠C + m ∠D = 180 Ssuapmpele-smideenitnatrey.rior angles are

m∠C + 60 = 180 Substitute. fromhesacmh 1sid1eg. mse_06cr_t06387h.asim11gmse_06cr_t06390.ai
m∠C = 120 Subtract 60

Chapter 6  Chapter Review 423

6-7  Polygons in the Coordinate Plane

Quick Review Exercises

To determine whether sides or diagonals are congruent, Determine whether △ABC is scalene, isosceles, or
use the Distance Formula. To determine the coordinate of equilateral.
the midpoint of a side, or whether the diagonals bisect each
other, use the Midpoint Formula. To determine whether 38. 2 y 39. y (3, 3)
opposite sides are parallel, or whether diagonals or sides are (؊1, 1) (0, 2)
perpendicular, use the Slope Formula. x

Example Ϫ2 O 2 O2 x
Ϫ2 (1, ؊1) 4
△XYZ has vertices X(1, 0), Y( −2, −4 ), and Z(4, −4 ).
Is △XYZ scalene, isosceles, or equilateral? (؊1, ؊2) (3, ؊2)

To find the lengths of the legs, use the Distance Formula. What is the most precise classification of the
quadrilateral?
XY = 2( - 2 - 1)2 + ( - 4 - 0)2 = 19 + 16 = 5
YZ = 2(4 - ( -2))2 + ( -4 - ( -4))2 = 136 + 0 = 6 40. G(2h,s5m), 1R(15g, m8),sAe(_-026,c1r2_)t,0D6(3-h95s2,m9.a)1i 1gmse_06cr_t06393.ai

XZ = 2(4 - 1)2 + ( - 4 - 0)2 = 19 + 16 = 5 41. F( -13, 7), I(1, 12), N(15, 7), E(1, -5)
Two side lengths are equal, so △XYZ is isosceles.
42. Q(4, 5), U(12, 14), A(20, 5), D(12, -4)

43. W( -11, 4), H( -9, 10), A(2, 10), T(4, 4)

6-8 and 6-9  Coordinate Geometry and Coordinate Proofs

Quick Review Exercises y x
F
When placing a figure in the coordinate plane, it is usually 44. In rhombus FLPS, the axes
helpful to place at least one side on an axis. Use variables form the diagonals. If L
when naming the coordinates of a figure in order to show SL = 2a and FP = 4b, S
that relationships are true for a general case. what are the coordinates of
the vertices?
P

Example Py Q 45. The figure at the right is a (؊b, c) yP
x parallelogram. Give the
Rectangle PQRS has length a and O coordinates of point P hsm11gmse_0x 6cr_t06395.ai
width 4b. The x-axis bisects PS and S R without using any new
QR. What are the coordinates of the variables. (؊a, 0) O
vertices?

Since the width of PQRS is 4b and the x-axis bisects PS and 46. Use coordinate geometry to prove that the

yQ-Rax, iasl,lstohePv=er(t0ic,e2sba)raen2dbSu=nit(s0,fr-om2b)th. Tehhxse-malex1nis1g. gtPhmSoifssePo_Qn0Rt6hScer_t06394.aqauki aitderiislaaterreacltafonrgmlee.d by connecting the midpoints of
is a, so Q = (a, 2b) and R = (a, -2b). hsm11gmse_06cr_t06396.ai

424 Chapter 6  Chapter Review

6 Chapter Test MathX

OLMathXL® for School
R SCHO Go to PowerGeometry.com
L®

FO

Do you know HOW? 16. Algebra  Determine the B 7x Ϫ 1 C
1. What is the sum of the interior angle measures of a values of the variables for
which ABCD is a 7x Ϫ 2 5x ϩ 2
polygon with 15 sides? parallelogram.
2. What is the measure of an exterior angle of a 25-gon?
Give the coordinates for points A 6x ϩ y D
Graph each quadrilateral ABCD. Then determine the
most precise name for it. S and T without using any new variables. Then find the
3. A(1, 2), B(11, 2), C(7, 5), D(4, 5)
4. A(3, -2), B(5, 4), C(3, 6), D(1, 4) midpoint and the slope of ST .

17. rectangle 18. phayrsam(lcl,e1dlo1) ggrmamseT_06ct_t05839.ai
T y (a, b)

5. A(1, -4), B(1, 1), C( -2, 2), D( -2, -3) x x
O S (b, 0)
Algebra  Find the values of the variables for each
quadrilateral. S (a, ؊b)

6. xЊ 7. 19. Prove that the diagonals of y C(a, a)
xЊ zЊ square ABCD are congruent. D(0, a)
yЊ
50Њ yЊ
57Њ
20. Shkestmch1t1wgomnosen_co0n6gcrtu_etn0t5840h.asmi 11gmse_06ct_t05841.ai
parallelograms ABCD and x

108.. h3h20yssЊmϩm211141xggϪmmx4Њ ss3eex _ϩ_y0202Њx66cctt__tt0055188913..3 35hh..zasasЊmimiy38Њ1160ЊЊ11xyЊggЊmmzxЊЊssee__0066cctt__tt0055883364..a Da2ii1o. tOOEaAychF..pCno aG sreeteuq≅Hndvcu-eetUEsaBartunretnDNeecrdg xmhel≅Deidtis hnE EaaeRWtGtetSrha≅iTteceAhoFtrNfHhiigge.Duicnr?oeaodbnwrdd..i tipohtnranahatrehtpaseelsemslizegdool1ieofvi1dfgeiAosgrnua3mcmruopsnnoeidit_nsit0tlisoo6Bnn(cagst,.._0)t05842.ai

60Њ 22. Writing  Explain why a square cannot be a kite.

Does the information help you A B 23. Error Analysis  Your classmate says, “If the diagonals
prove that ABCD is a of a quadrilateral intersect to form four congruent
parallelogram? Explain.
hsm11gmse_06ct_t05838.ai triangles, then the quadrilateral is always a

12. AhCsmbi1se1cgtsmBsDe._06ct_t05837.ai D C parallelogram.” How would you correct this
statement? Explain your answer.

13. AB ≅ DC, AB } DC 24. Reasoning  PQRS has vertices P(0, 0), Q(4, 2), and

14. AB ≅ DC , BC ≅ AD ≅ ∠CDAhsm11gmse_06ct_t05844.atESih(x4ep,pla-oi2nss).i.bItles diagonals intersect at H(4, 0). What are
15. ∠DAB ≅ ∠BCD , ∠ABC coordinates of R for PQRS to be a kite?

Chapter 6  Chapter Test 425

6 Common Core Cumulative ASSESSMENT
Standards Review

Some questions on tests ask In rhombus QRST, what is m∠RST ? TIP 1
you to find an angle measure QT
in a figure. Read the sample The diagonal QS forms
question at the right. Then 4.5 cm △Q RS.
follow the tips to answer it.
9 cm Think It Through
TIP 2 P
RS = 9 cm because a
Use the properties of a RS rhombus has four congruent
rhombus to find RS and sides. The diagonals of a
QS. 60 120
#rhombus bisect each other, so
90 150
QS = 2 4.5 = 9 cm.
hsm11gmse_06cu_t06098.ai Since QR = RS = QS, △QRS
is equilateral, and thus
equiangular. So m∠RSQ = 60.
Since the diagonals of a

#rhombus also bisect the angles,

m∠RST = 2 60 = 120. The
correct answer is C.

LVVeooscsacoabnubluarlayry 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. Choose the answer on your paper.
correct term to complete each sentence.
1. Which quadrilateral must have congruent diagonals?
A. Two lines that intersect to form right angles are
(parallel, perpendicular) lines. kite parallelogram

B. A quadrilateral with two pairs of adjacent sides rectangle rhombus
congruent and no opposite sides congruent is
called a (rhombus, kite). 2. STUV is a parallelogram. What are the coordinates of

C. The (circumcenter, incenter) of a triangle is point U?
the point of concurrency of the perpendicular
bisectors of the sides of the triangle. (x, y) y

D. Two angles are (complementary, (x + z, y) T(0, y) U(?, ?)
supplementary) angles if the sum of their
measures is 90. (y, z)

E. In a plane, two lines that never intersect are (z, y)
(parallel, skew) lines.
O S(x, 0) x
V(x ؉ z, 0)

hsm11gmse_06cu_t06100.ai

426 Chapter 6  Common Core Cumulative Standards Review

3. Which list could represent the lengths of the sides of a 8. For which value of x are lines g and h parallel?
triangle?

7 cm, 10 cm, 25 cm
(2x ϩ 10)Њ g
4 in., 6 in., 10 in. (5x Ϫ 5)Њ

1 ft, 2 ft, 4 ft h

3 m, 5 m, 7 m 12 18
15 25
4. Which quadrilateral CANNOT contain four right

angles? 9. In △GHJ, GH ≅ HJ . Using the indirect proof method,

square trapezoid you attemhpstmto1p1rgovme sthea_t0∠6Gcua_ntd0∠61J 0a4re.ariight angles.

rhombus rectangle Which theorem will contradict this claim?
Triangle Angle-Sum Theorem
5. What is the circumcenter of △ABC with vertices
Side-Angle-Side Theorem
A( -7, 0), B( -3, 8), and C( -3, 0)?
Converse of the Isosceles Triangle Theorem
( -7, -3) ( -4, 3)

( -5, 4) ( -3, 4) Angle-Angle-Side Theorem

6. ABCD is a rhombus. To prove that the diagonals 10. Which angles could an obtuse triangle have?
of a rhombus are perpendicular, which pair of
angles below must you prove congruent by using two right angles two obtuse angles
corresponding parts of congruent triangles?
two acute angles two vertical angles

A B 1 1. What values of x and y make the quadrilateral below a
parallelogram?
E


yϩ1 5x Ϫ 6

DC 3x Ϫ 2 6y Ϫ 4

∠AEB and ∠DEC x = 2, y = 1 x = 1, y = 2
∠AEB and ∠AED x = 3, y = 5
∠BEC and ∠AED x = 2, y = 9
7
∠DhAsBma1n1dg∠mABseC_06cu_t06102.ai
12. Whichhissmthe11mgomst sveal_id06cocnuc_lut0si6o1n0b5as.aedi on the
7. FGHJ is a quadrilateral. If at least one pair of opposite
angles in quadrilateral FGHJ is congruent, which statements below?
statement is false?
If a triangle is equilateral, then it is isosceles. △ABC is
Quadrilateral FGHJ is a trapezoid. not equilateral.
Quadrilateral FGHJ is a rhombus.
Quadrilateral FGHJ is a kite. △ABC is not isosceles.
Quadrilateral FGHJ is a parallelogram.
△ABC is isosceles.

△ABC may or may not be isosceles.

△ABC is equilateral.

Chapter 6  Common Core Cumulative Standards Review 427

Constructed Response 1 9. The pattern of a soccer
ball contains regular
1 3. What is m∠1 in the figure below? hexagons and regular
pentagons. The figure
31Њ at the right shows what
a section of the pattern
38Њ would look like on a flat
surface. Use the fact that
1 there are 360° in a circle
to explain why there
69Њ are gaps between the hexagons.

14. ∠ABE and ∠CBD are vertical angles and both are Extended Response
complementary with ∠FGH. If m∠ABE = (3x - 1), hsm11gmse_06cu_t06113.ai
and m∠FGH = 4x, what is m∠CBD?
20. Prove that GHIJ is a rhombus.
15. What is thhe svmalu1e1ogfmx isnet_h0e6kcitue_bte0lo6w1?08 .ai
6y

4
22Њ xЊ 2H I

1 6. A 3-ft-wide walkway is placed around an animal Ϫ2 2 x
exhibit at the zoo. The exhibit is rectangular in shape Ϫ2 6
J
and has lenhgtshm1151ftgamndsew_id0t6hc8uf_t.tW06h1at0i9s.tahie area, in
G Ϫ4
square feet, of the walkway around the exhibit?
17. The outer walls of the Pentagon in Arlington, Virginia, 21. A parallelogram has vertices L( -2, 5), M(3, 3), and
N(1, 0). What are possible coordinates for its fourth
are formed by two regular pentagons, as shown below. vertex? Explain.
What is the value of x?
22. Jim is cleagrienog mou1t 2tr_esees _ancdcsp_lacn0t6incgsrg_rat0ss1t.oaienlarge

xЊ his backyard. He plans to double both the width and
the length of his current rectangular backyard, as
shown below.

New

18. What are the possible values for n to make ABC a valid Current 2w
triangle? Show your work. w

hsm11gCmse_06cu_t06110.ainϩ1 ഞ 2ഞ

2n a. How will the area of Jim’s new backyard compare
to the area of his current backyard?
A 5n Ϫ 4 B
b. Will it take Jim twice as long to mow his new

bJhiamscmkmy1ao1rwdgsmtahtastneh_he0iss6aomclude_brtaa0tce6kfy1oa1rr5db?.oatEihxpyalaridns..)(Assume

428 Chapter 6  Common Core Cumulative Standards Review

hsm11gmse_06cu_t06112.ai

CHAPTER

Get Ready! 7

Lesson 3-2 Properties of Parallel Lines

Use the diagram at the right. Find the measure of each angle. 2 a
Justify your answer.

1. ∠1 2. ∠2 3. ∠3 4. ∠4 4 1 110Њ b
3
Lesson 4-1 Naming Congruent Parts

△PAC ≅ △DHL. Complete each congruence statement.

5. PC ≅ ? 6. ∠H ≅ ? 7. ∠PCA ≅ ? 8. △HDL ≅ ?

Lessons 4-2 Triangle Congruence hsm11gmse_07co_t05109

and 4-3 Write a congruence statement for each pair of triangles. Explain why the
triangles are congruent.

9. K L 10. A 11. U
D

NB
HB

P MC G

E

Lesson 5-1 Midsegments of Triangles A

U1s2e. thIhfsBemCd1i=a1g1gr2am,mtshaeet_nt0hB7eFcr=oig_ht?t0f5oarn1dE1hx0sDemErc1=is1eg?sm1.2s–e1_30. 7co_t0511h1sm11Dgmse_07cEo_Bt05112
CF
13. If EF = 4.7, then AD = ? and AC = ? .

Looking Ahead Vocabulary hsm11gmse_07co_t05113

14. An artist sketches a person. She is careful to draw the different parts of the person’s
body in proportion. What does proportion mean in this situation?

15. Siblings often look similar to each other. How might two geometric figures be
similar?

16. A road map has a scale on it that tells you how many miles are equivalent to
a distance of 1 inch on the map. How would you use the scale to estimate the
distance between two cities on the map?

Chapter 7  Similarity 429

7CHAPTER Similarity

Download videos VIDEO Chapter Preview 1 Similarity
connecting math Essential Question  How do you use
to your world.. 7-1 Ratios and Proportions proportions to find side lengths in similar
7-2 Similar Polygons polygons?
Interactive! 7-3 Proving Triangles Similar
Vary numbers, 7-4 Similarity in Right Triangles 2 Reasoning and Proof
graphs, and figures 7-5 Proportions in Triangles Essential Question  How do you show
to explore math two triangles are similar?
concepts.. ICYNAM
ACT I V I TI 3 Visualization
D Essential Question  How do you identify
ES corresponding parts of similar triangles?

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

Math definitions VOC ABUL ARY Vocabulary DOMAINS
in English and • Similarity, Right Triangles, and Trigonometry
Spanish English/Spanish Vocabulary Audio Online:
• Expressing Geometric Properties with
English Spanish Equations

extremes of a valores extremos de una • Mathematical Practice: Construct viable
proportion, p. 434 proporción arguments
Online access
to stepped-out geometric mean, p. 462 media geométrica
problems aligned
to Common Core indirect measurement, p. 454 medición indirecta
Get and view
your assignments NLINE means of a proportion, p. 434 valores medios de una
online. ME WO proporción

O proportion, p. 434 proporción
RK
HO ratio, p. 432 razón

scale drawing, p. 443 dibujo a escala

Extra practice scale factor, p. 440 factor de escala
and review
online similar, p. 440 semejante

similar polygons, p. 440 polígonos semejantes

Virtual NerdTM
tutorials with
built-in support

PERFORMANCE TASK

Common Core Performance Task

Adjusting a Graphing Calculator Window

Lillian graphs the functions y = 2x + 3 and y = - 1 x + 1 on her graphing
2
calculator. She knows the lines are perpendicular because the product of the

slopes of the lines is -1. However, the lines do not appear to be perpendicular on

the screen.

Plot1 Plot2 Plot3 WINDOW  
\Y1 = 2X+3 Xmin = –10
\Y2 = –(1/2)X+1 Xmax = 10
\Y3 = Xscl = 1
\Y4 = Ymin = –10
\Y5 = Ymax = 10
\Y6 = Yscl = 1
\Y7 =
TXres = 1
 

The WINDOW screen shows the intervals of the x-axis (from Xmin to Xmax) and
the y-axis (from Ymin to Ymax) that are visible in the graph. Lillian wants to adjust
the values of Xmin and Xmax that she entered so that the graph of the lines is not
distorted. She measures her calculator’s rectangular screen and finds the width is
5.1 cm and the height is 3.4 cm.

Task Description

Determine the values Lillian should enter for Xmin and Xmax so that the graph is
not distorted.

Connecting the Task to the Math Practices MATHEMATICAL

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

• You’ll use ratios to help you understand the problem. (MP 1)

• You’ll determine whether two rectangles are similar and explain your
reasoning. (MP 3)

Chapter 7  Similarity 431

7-1 Ratios and Proportions MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To write ratios and solve proportions Prepares for GM-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.iteria for
tsriimanilagrleitsy tcoristeorlviae fporrotbrilaenmgsleasntdotsooplvreovperoreblaetmiosnasnhdiptso
ipnrogveeomreelattriocnfisghuipressi.n geometric figures.

MP 1, MP 3, MP 4, MP 6, MP 7

The table at the right gives the wins and losses Year Wins Losses
of a baseball team. In which year(s) did the team 1890 60 24
have the best record? Explain. 110
1930 44

The year the team 1970 110 52
had the most wins 2010 108 54
is not necessarily
the year in which
it had the best
record.

MATHEMATICAL In the Solve It, you compared two quantities for four years.
PRACTICES
hsm11gmse_0701_t13029
Essential Understanding  You can write a ratio to compare two quantities.

Lesson A ratio is a comparison of two quantities by division. You can write the ratio of two
a
Vocabulary numbers a and b, where b ≠ 0, in three ways: b , a : b, and a to b. You usually express a
• ratio
and b in the same unit and write the ratio in simplest form.

• extended ratio

• proportion

• extremes Problem 1 Writing a Ratio
• means

• Cross Products Bonsai Trees  The bonsai bald cypress tree is a small
Property version of a full-size tree. A Florida bald cypress tree

called the Senator stands 118 ft tall. What is the ratio of

How can you write the height of the bonsai to the height of the Senator? 15 in.

the heights using the Express both heights in the same unit. To convert 118 ft
same unit? 12 in.
You can convert the to inches, multiply by the conversion factor 1 ft .
height of the Senator to
inches or the height of 118 ft 5 118 ft ? 12 in. 5 (118 ? 12) in. 5 1416 in.
the bonsai tree to feet. 1 1 ft

Write the ratio as a fraction in simplest form.

height of bonsai S 15 in. ϭ 15 in. ϭ (3 ؒ 5) in. ϭ 5
height of Senator S 118 ft 1416 in. (3 ؒ 472) in. 472

The ratio of the height of the bonsai to the height of the Senator is 5 or 5 : 472.
472

Got It? 1. A bonsai tree is 18 in. wide and stands 2 ft tall.
What is the ratio of the width of the bonsai to its height?

432 Chapter 7  Similarity

Problem 2 Dividing a Quantity Into a Given Ratio

Fundraising  Members of the school band are buying pots of tulips and pots of

daffodils to sell at their fundraiser. They plan to buy 120 pots of flowers. The ratio
number of tulip pots 2
number of daffodil pots will be 3 . How many pots of each type of flower should they buy?

How do you write

expressions for the If the ratio number of tulip pots is 2 , it Let 2x = the number of tulip pots.
number of daffodil pots 3 Let 3x = the number of daffodil pots.
numbers of pots?
must be in the form 2x . 2x + 3x = 120
Multiply the numerator 3x 5x = 120
x = 24
2 and the denominator 3
2x 2 2x = 2(24) = 48
by the factor x. 3x = 3 . The total number of flower pots is 120. 3x = 3(24) = 72
Use this fact to write an equation. Then
solve for x. The band members should buy 48 tulip
pots and 72 daffodil pots.
Substitute 24 for x in the expressions for
the numbers of pots.

Write the answer in words.

Got It? 2. The measures of two supplementary angles are in the ratio 1 : 4. What are
the measures of the angles?

An extended ratio compares three (or more) numbers. In the extended ratio a : b : c,
the ratio of the first two numbers is a : b, the ratio of the last two numbers is b : c, and
the ratio of the first and last numbers is a : c.

Problem 3 Using an Extended Ratio

The lengths of the sides of a triangle are in the extended ratio 3 : 5 : 6. The perimeter
of the triangle is 98 in. What is the length of the longest side?

Sketch the triangle. Use the extended ratio to label the sides with 3x 5x
expressions for their lengths.
How do you use
the solution of the 3x + 5x + 6x = 98 The perimeter is 98 in. 6x
equation to answer
the question? 14x = 98 Simplify.

Substitute the value for x x=7 Divide each side by 14.

in the expression for the The expression that represents the length of the longest side is 6x. 6(7)h=sm421, 1sogtmhese_0701_t05114
length of the longest side is 42 in.
length of the longest side.

Got It? 3. The lengths of the sides of a triangle are in the extended ratio 4 : 7 : 9. The
perimeter is 60 cm. What are the lengths of the sides?

Lesson 7-1  Ratios and Proportions 433

Essential Understanding  If two ratios are equivalent, you can write an
equation stating that the ratios are equal. If the equation contains a variable, you can
solve the equation to find the value of the variable.

An equation that states that two ratios are equal is called a proportion. The first and last
numbers in a proportion are the extremes. The middle two numbers are the means.

T extremes T extremes S 2 ϭ 4
2Ϻ3 ϭ 4Ϻ6 means S 3 6

mc eancs

Key Concept  Cross Products Property

Words Symbols Example
In a proportion, the product a c # #2 4
of the extremes equals the If b = d , where b ≠ 0 and 3 = 6
product of the means.
d ≠ 0, then ad = bc. 2 6=3 4
12 = 12

Here’s Why It Works  Begin with a = c , where b ≠ 0 and d ≠ 0.
# # bd b d

a = c bd Multiply each side of the proportion by bd.
# # b d Divide the common factors.

bd a = c b1d
1 b d

ad = bc Simplify.

Problem 4 Solving a Proportion

Algebra  What is the solution of each proportion?

A 6 = 5 B y + 4 = y
x 4 9 3
6(4) = 5x 3(y + 4) = 9y
Cross Products Property
Does the solution 24 = 5x Simplify. 3y + 12 = 9y

check? x = 24 Solve for the variable. 12 = 6y
5
# #6≟ 5 y = 2
4
245
6 ≟ 24 5 The solution is 24 or 4.8. The solution is 2.
4 5 5

24 = 24 ✓

Got It? 4. What is the solution of each proportion?

a. 92 = 1a4 b. 15 1 = 3
m+ m

434 Chapter 7  Similarity

Using the Properties of Equality, you can rewrite proportions in equivalent forms.

Key Concept  Properties of Proportions

a, b, c, and d do not equal zero.

Property How to Apply It
a c b dc .
(1) b = d is equivalent to a = Write the reciprocal of each ratio.

2 ϭ 4 becomes 3 ϭ 6 .
3 6 2 4

(2) a = c is equivalent to a = b . Switch the means.
b d c d
2 4 2 3
3 ϭ 6 becomes 4 ϭ 6 .

(3) a = c is equivalent to a + b = c + d. In each ratio, add the denominator to the
b d b d
numerator.
2 + 3 4 + 6.
2 = 4 becomes 3 = 6
3 6

Problem 5 Writing Equivalent Proportions

In the diagram, x = 7y . What ratio completes the equivalent x
6 6
x ■
How do you decide proportion y = ■ ? Justify your answer.
which property of
proportions applies? Method 1 Method 2 y7
Look at how the y
positions of the known x = 7 x = y
parts of the incomplete 6 6 7
proportion relate to their
positions in the original x = 6 Property of Proportions (2) 7x = 6y Cross Products Property
proportion. y 7

7x = 6y To solve for yx, divihdesmeac1h1sgidme bsye7_y.0701_t05115
7y 7y
x 6
y = 7 Simplify.

The ratio that completes the proportion is 67.

Got It? 5. For parts (a) and (b), use the proportion x = y . What ratio completes the
6 7
equivalent proportion? Justify your answer.

a. 6x = ■■
b. ■■ = y +7 7

c. tRoe6xas=on7y.ing  Explain why x -6 6 = y -7 7 is an equivalent proportion

Lesson 7-1  Ratios and Proportions 435

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

1. To the nearest millimeter, a cell phone is 84 mm long PRACTICES
and 46 mm wide. What is the ratio of the width to the
length? 5. Vocabulary  What is the difference between a ratio

and a proportion?

2. Two angle measures are in the ratio 5 : 9. Write 6. Open-Ended  The lengths of the sides of a triangle are
expressions for the two angle measures in terms of in the extended ratio 3 : 6 : 7. What are two possible
the variable x. sets of side lengths, in inches, for the triangle?

3. What is the solution of the proportion 20 = 5 ? 7. Error Analysis  What is the error 7 = 4
z 3 in the solution of the proportion 3 x
shown at the right?
4. For a = 13 complete each equivalent proportion. 28 = 3x
7 b 8. What is a proportion that has
means 6 and 18 and extremes
a. ■a = 7 b. a - 7 = ■■ c. 7 = ■ 9 and 12? 28 = x
■ 7 a ■ 3

Practice and Problem-Solving Exercises MATHEMATICAL hsm11gmse_0701_t05116

PRACTICES

A Practice Write the ratio of the first measurement to the second measurement. See Problem 1.

9. length of a tennis racket: 2 ft 4 in. 10. height of a table tennis net: 6 in.

length of a table tennis paddle: 10 in. height of a tennis net: 3 ft

11. diameter of a table tennis ball: 40 mm 12. length of a tennis court: 26 yd

diameter of a tennis ball: 6.8 cm length of a table tennis table: 9 ft

13. Baseball  A baseball team played 154 regular season games. The ratio of the See Problem 2.
See Problem 3.
number of games they won to the number of games they lost was 5 . How many
2
games did they win? How many games did they lose?

14. The measures of two supplementary angles are in the ratio 5 : 7. What is the
measure of the larger angle?

15. The lengths of the sides of a triangle are in the extended ratio 6 : 7 : 9. The
perimeter of the triangle is 88 cm. What are the lengths of the sides?

16. The measures of the angles of a triangle are in the extended ratio 4 : 3 : 2. What is
the measure of the largest angle?

Algebra  Solve each proportion. See Problem 4.

17. 13 = 1x2 18. 9 = 3 19. 4 = 5 20. y = 15 21. 9 = 12
5 x x 9 10 25 24 n

22. 1114 = 2b1 23. 3 = 6 3 24. y + 7 = 8 25. x 5 = 1x0 26. n + 4 = n
5 + 9 5 - 8 4
x 3

436 Chapter 7  Similarity

In the diagram, a = 3 . Complete each statement. Justify your answer. See Problem 5.
b 4 E
27. ab = ■■ 28. 4a = ■ ■
29. ■ = b 4
4
30. ■■ = 74 31. a + b ■ b 4 3
b = ■ 32. ■ = ■
a
B Apply b 8y

Coordinate Geometry  Use the graph. Write each

ratio in simplest form.

33. BADC 34. AE 35. slope of EB 36. slope of ED 6
EC The of a rectangle is 150
in.2. hThsemr1at1iogmse_07014_t05117
37. Think About a Plan  area

of the length to the width is 3 : 2. Find the length and the width.

• What is the formula for the area of a rectangle? AO x
• How can you use the given ratio to write expressions for the BC6D

length and width?

Art To draw a face, you can sketch the head as an oval and then hsm11gmse_0701_t05118
lightly draw horizontal lines to help locate the eyes, nose, and
mouth. You can use the extended ratios shown in the diagrams to
help you place the lines for an adult’s face or for a baby’s face.

38. If AE = 72 cm in the diagram, find AB, BC, CD, and DE.

39. You draw a baby’s head as an oval that is 21 in. from top to bottom.
a. How far from the top should you place the line for the eyes?
b. Suppose you decide to make the head an adult’s head. How far

up should you move the line for the eyes?

Algebra Solve each proportion.

40. 7y 1- 5 = 92y 41. 4a + 1 = 2a 42. x 5 2 = x 3 1 43. 2b - 1 = b-2
7 3 + + 4 12

44. The ratio of the length to the width of a rectangle is 9 : 4. The width of the rectangle
is 52 mm. Write and solve a proportion to find the length.

45. Open-Ended  Draw a quadrilateral that satisfies this condition: The measures of the
consecutive angles are in the extended ratio 4 : 5 : 4 : 7.

46. Reasoning   The means of a proportion are 4 and 15. List all possible pairs of
positive integers that could be the extremes of the proportion.

47. Writing  Describe how to use the Cross Products Property to determine whether

10 = 16 is a true proportion.
26 42

48. Reasoning  Explain how to use two different properties of proportions to change

the proportion 3 = 12 into the proportion 12 = 146 .
4 16 3

Lesson 7-1  Ratios and Proportions 437

Complete each statement. Justify your answer.

49. If 4m = 9n, then mn = ■ . 50. If 30 = 1r8, then t = ■■. 51. If a + 5 = b + 2, then a = ■■.
■ t r 5 2 5

52. If ab = dc , then ac ++ db = ■ . 53. If a = c , then a + c = ■ . 54. If a = c , then a + 2b = ■ .
■ b d b + d ■ b d b ■

C Challenge Algebra  Use properties of equality to justify each property of proportions.

55. ab = dc is equivalent to b = d . 56. a = c is equivalent to a = b .
a c b d c d

57. ab = dc is equivalent to a + b = c + d.
b d

Algebra  Solve each proportion for the variable(s).

58. x -3 3 = x +2 2 59. 3 - 4x = 2 1 3x 60. x = x + 10 = 4x
1 + 5x + 6 18 y

PERFORMANCE TASK

Apply What You’ve Learned MATHEMATICAL

PRACTICES

MP 1

Look back at the information on page 431 about the dimensions of Lillian’s

graphing calculator screen. Select all of the following that are true. Explain your

reasoning.

A. The perimeter of the calculator screen is 10 cm.

B. The ratio of the width of the screen to the height of the screen is 3 : 2.

C. The ratio of the width of the screen to the height of the screen is 15 : 9.

D. The ratio of the width of the screen to the height of the screen is the same whether
you measure the dimensions in centimeters or millimeters.

E. A rectangle with a width of 2 ft and a height of 16 in. has the same ratio of width to
height as Lillian’s calculator screen.

F. An extended ratio that compares the dimensions of the four sides of Lillian’s
calculator screen is 3 : 2 : 3 : 2.

G. The extended ratio that compares the dimensions of the four sides of Lillian’s
calculator screen is 2 : 1 : 2 : 1.

438 Chapter 7  Similarity

Algebra Review Solving Quadratic MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 7-2 Equations Reviews AM-ACFESD..9A1.21. AC-rCeEatDe.1eq.1u aCtiroenasteand
ienqeuqautaiolintiseasnind oinneeqvuaarliatibelseiannodnuesveatrhiaebmletaond
usoselvethpermobtloemsosl.ve problems.

Equations in the form ax2 + bx + c = 0, where a ≠ 0, are quadratic equations in
standard form. You can solve some quadratic equations in standard form by factoring

and using the Zero-Product Property:

If ab = 0, then a = 0 or b = 0.

You can solve all quadratic equations in standard form using the quadratic formula:

If ax2 + bx + c = 0, where a ≠ 0, then x = -b { 2b2 - 4ac.
2a

Example

Algebra  Solve for x. For irrational solutions, give both the exact answer and the
answer rounded to the nearest hundredth.

A 7x2 + 6x - 1 = 0 The equation is in standard form.
(7x - 1)(x + 1) = 0 Factor.
Use the Zero-Product Property.
7x - 1 = 0 or x + 1 = 0 Solve for x.
x = 71 or   x = - 1
The equation is in standard form.
B - 3x2 - 5x + 1 = 0 Identify a, b, and c.

a = -3, b = -5, c = 1 Substitute in the quadratic formula.

x = - ( - 5) { 2( - 5)2 - 4( - 3)(1) Simplify.
2( - 3) Write the two solutions separately.
5 { 137 Use a calculator and round to the nearest hundredth.
x = -6

x = - 5 + 6137 or x = - 5 - 137
6

x ≈ -1.85 or x ≈ 0.18

Exercises

Algebra  Solve for x. For irrational solutions, give both the exact answer and the
answer rounded to the nearest hundredth.

1. x2 + 5x - 14 = 0 2. 4x2 - 13x + 3 = 0 3. 2x2 + 7x + 3 = 0
4. 5x2 + 2x - 2 = 0 5. 2x2 - 10x + 11 = 0 6. 8x2 - 2x - 3 = 0
7. 2x2 + 3x - 20 = 0 8. x2 - x - 210 = 0 9. x2 - 4x = 0
10. x2 - 25 = 0 11. 6x2 + 10x = 5 12. 1 = 2x2 - 6x

Algebra Review  Solving Quadratic Equations 439

7-2 Similar Polygons CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Objective To identify and apply similar polygons MG-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.isteimriailafroitrytrciraintegrlieasftoor
tsroialvnegplersobtolesmoslvaenpdrotoblpermosveanredlattoiopnrsohviepsreilnatgioeonmsheiptrsic
ifnigguereosm. etric figures.

MP 1, MP 3, MP 4, MP 6

Can you use what A movie theater screen is in the shape of
you’ve learned a rectangle 45 ft wide by 25 ft high.
before about ratios Which of the TV screen formats at the
to help you solve right do you think would show the most
this problem? complete scene from a movie shown on the
theater screen? Explain.

MATHEMATICAL

PRACTICES Similar figures have the same shape but not necessarily the same size. You can
abbreviate is similar to with the symbol ∼ .

Essential Understanding  You can use ratios and proportions to decide whether
two polygons are similar and to find unknown side lengths of similar figures.

Lesson Key Concept  Similar Polygons

Vocabulary Define Diagram Symbols
• similar figures Two polygons are ABCD ∼ GHIJ
• similar polygons similar polygons if     ∠A ≅ ∠G
• extended corresponding angles are
congruent and if the BC H I      ∠B ≅ ∠H
proportion lengths of corresponding
• scale factor sides are proportional.  ∠C ≅ ∠I
• scale drawing
• scale ∠D ≅ ∠J

AD AB = BC = CD = AD
G GH HI IJ GJ
J

You write a similarity statement with corresponding vertices in order, just as you write

a congruence statement. When three or more ratios are equal, you can write an

extended proportion. The proporthiosnmGA1HB1=gmBHCIs=e_CI0DJ 7=02AGD_J ti0s 5an1 2e1xtended proportion.

A scale factor is the ratio of corresponding linear measurements B

of two similar figures. The ratio of the lengths of corresponding 15 20 Y
sides BC and YZ, or more simply stated, the ratio of 68
25isoBYrCZ5 =: 228.0 25 .
corresponding sides, = So the scale factor A 25 C X 10 Z
of △ABC to △XYZ is
△ABC ∼ △XYZ

440 Chapter 7  Similarity

Problem 1 Understanding Similarity

△MNP ∼ △SRT M PR

How can you use the A What are the pairs of congruent angles? N
TS
s imilarity statement ∠M ≅ ∠S, ∠N ≅ ∠R, and ∠P ≅ ∠T
to write ratios of
corresponding sides? B What is the extended proportion for the ratios of
corresponding sides?
Use the order of the

ss tidaetesminentht.eMsiNmilarity MN = NP = MP
SR RT ST

scc ooorrMrrSeeRNsspp ioosnnaddrisanttgoiosSiodRfe,s. Got It? 1. bDa..E WFWGhhaa∼ttaiHsre JtKhtheLe .epxtaeinrsdoefdcpornogprouretniotnanfogrleths?e ratios of the lengths of
hsm11gmse_0702_t05623
corresponding sides?

Problem 2 Determining Similarity

Are the polygons similar? If they are, write a similarity statement
and give the scale factor.

A JKLM and TUVW J 12 K T6U
Step 1 Identify pairs of congruent angles. 6 6
∠J ≅ ∠T, ∠K ≅ ∠U, ∠L ≅ ∠V, and ∠M ≅ ∠W M
W 16
How do you identify Step 2 Compare the ratios of corresponding sides. 24 24 14
corresponding sides?
The included side JK = 12 = 2 KL = 24 = 3 V
between a pair of TU 6 1 UV 16 2
angles of one polygon L
corresponds to the LM = 24 = 12 JM = 6 = 1
included side between VW 14 7 TW 6 1
the corresponding pair
of congruent angles of Corresponding sides are not proportional, so the polygons are not similar.
another polygon.
B △ABC and △EFD Ahsm1211gmB seE_07022_0t05624F
Step 1 Identify pairs of congruent angles. 8 16
∠A ≅ ∠D, ∠B ≅ ∠E, and ∠C ≅ ∠F 15 10
CD

Step 2 Compare the ratios of corresponding sides.

AB = 12 = 54 BC = 16 = 54 AC = 8 = 4
DE 15 EF 20 DF 10 5

Yes; △ABC ∼ △DEF and the scale factor is 4 or 4 : 5. hsm11gmse_0702_t05625
5

Got It? 2. Are the polygons similar? If they are, write a similarity statement and give

the scale factor.

a. K 10 L W 20 X b. A 18 B R 9 S6
12 18 18
C
15 15 E 9 6T
12 V 9U

N MZ Y D

Lesson 7-2  Similar Polygons 441

Problem 3 Using Similar Polygons

Can you rely on the Algebra  ABCD ∼ EFGD. What is the value of x?

d iagram alone to set 4.5 7.2 A5 B
Nu po, the proportion? 5 11.25
you need to use the 9 E yF 7.5
6 x
similarity statement to FG = ED Corresponding sides of similar D
BC AD polygons are proportional.
identify corresponding

sides in order to write x 96 Substitute. GC
7.5
ratios that are equal. =

9x = 45 Cross Products Property

x = 5 Divide each side by 9.

The value of x is 5. The correct answer is B. hsm11gmse_0702_t05628.ai

Got It? 3. Use the diagram in Problem 3. What is the value of y?

Problem 4 Using Similarity

You can’t solve the Design  Your class is making a rectangular poster for a rally. The poster’s design is
problem until you know 6 in. high by 10 in. wide. The space allowed for the poster is 4 ft high by 8 ft wide.
which dimension fills the What are the dimensions of the largest poster that will fit in the space?
space first.
Step 1 Determine whether the height or width will fill the space first.

Height: 4 ft = 48 in. Width: 8 ft = 96 in.
48 in. , 6 in. = 8 96 in. , 10 in. = 9.6

The design can be enlarged at most 8 times.

Step 2 The greatest height is 48 in., so find the width.

6 = 10 Corresponding sides of similar polygons are proportional.
48 x

6x = 480 Cross Products Property

x = 80 Divide each side by 6.

The largest poster is 48 in. by 80 in. or 4 ft by 632 ft.

Got It? 4. Use the same poster design in Problem 4. What are the
dimensions of the largest complete poster that will fit in a space
3 ft high by 4 ft wide?

442 Chapter 7  Similarity

In a scale drawing, all lengths are proportional to their corresponding actual lengths.
The scale is the ratio that compares each length in the scale drawing to the actual
length. The lengths used in a scale can be in different units. For example, a scale might
be written as 1 cm to 50 km, 1 in. = 100 mi, or 1 in. : 10 ft.

You can use proportions to find the actual dimensions represented in a scale drawing.

Problem 5 Using a Scale Drawing STEM
Design  The diagram shows a scale drawing of the Golden Gate Bridge
in San Francisco. The distance between the two towers is the main
span. What is the actual length of the main span of the bridge?

The length of the main span in the scale drawing is 6.4 cm. Let s represent

the main span of the bridge. Use the scale to set up a proportion.

Why is it helpful 1 = 6s.4 length in drawing (cm)
to use a scale in 200 actual length (m)
different units?
s = 1280 Cross Products Property
1 cm : 200 m in the

same units would be The actual length of the main span of the bridge is 1280 m.
1 cm : 20,000 cm. When
is soelvainsigerthtoe wproorbklewmit,h2100 Got It? 5. a. Use the scale drawing in Problem 5. What is the actual height of the towers
t han 20,1000. b. aRbeoavseonthinegr oTahdewSapya?ce Needle in Seattle is 605 ft tall. A classmate wants
to make a scale drawing of the Space Needle on an 821 in.–by-11 in. sheet
of paper. He decides to use the scale 1 in. = 50 ft. Is this a reasonable

scale? Explain.

Lesson 7-2  Similar Polygons 443

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

PRACTICES

JDRT ∼ WHYX . Complete each statement. 5. Vocabulary  What does the scale on a scale drawing
indicate?
1. ∠D ≅ ? 2. RT = ■
YX WX 6. Error Analysis  The polygons VT
at the right are similar. Which NP
3. Are the polygons similar? If they are, write a similarity similarity statement is not U QR
statement and give the scale factor. correct? Explain.

D 18 16 P 12 L A.  TRUV ∼ NPQU
24 E 8 B.  RUVT ∼ QUNP

12 R 16 Q 7. Reasoning  Is similarity reflexive? Transitive?

H 24 G Symmetric? Justify your reasoning. hsm11gmse_0702_t05631.ai

4. △FGH ∼ △MNP. What is the value of x? 8. The triangles at the right are B
similar. What are three similarity
F P 10 N statements for the triangles?

1h5sm11gm1s2e_0702_t05x629.ai P

M S
H 20 G AR

Prahcstmic1e1gamnsde_0P7r0o2b_tl0e5m63-0S.aoi lving Exercises MATHEMATICAL hsm11gmse_0702_t05632.ai

PRACTICES See Problem 1.

A Practice List the pairs of congruent angles and the extended proportion that
relates the corresponding sides for the similar polygons.

9. RSTV ∼ DEFG 10. △CAB ∼ △WVT 11. KLMNP ∼ HGFDC

R D A B W K L H G
S E C

T P MC F
G
V F TV ND

Determine whether the polygons are similar. If so, write a similarity See Problem 2.

statement and give the scale factor. If not, explain.

12. hBsm11gmse_07E02_t056 33.1a3i . hsm4 1B1g4mse6_07E02_6t056 34.1a4i . hKsm11gmseR_070125_t05P635.ai

9 AF 34 17 8
9 C 15 15 4 C 4D 6 G 6
16 Q
9
A F 15 D J 30 L

hsm11gmse_0702_t05641.ai hsm11gmse_0702_t05636.ai hsm11gmse_0702_t05637.ai

444 Chapter 7  Similarity

15. M A R 8 U 16. H 14 17. J
I E
E N
18 15 21.3 25.4 63Њ 20
12 9 G 15 43Њ 70Њ
B 80Њ LA
H 10 T E L T L 24 18 24 R
S

Algebra  The polygons are similar. Find the value of each variable. See Problem 3.

18. 8 19. hs1m2 11gm9se_07x026_t0635.65 40.2a0i. h1s5m11xgmse_307702_t3005638.ai
hsm11xgmse_0702_t05639.ai
z y 25.5
y 5 y
6 10

STEM 21. Web Page Design  The space allowed for the mascot on a school’s Web page is See Problem 4.

120 pixels wide by 90 pixels high. Its digital image is 500 pixels wide by 375 pixels

hhsigmh1.1Wgmhaset_is07th0e2_lat1rg3e1s4t1image of thehmsmas1co1tgtmhastew_il0l 7fi0t o2n_tth0e5W64e4b.apai geh?sm11gmse_0702_t05645.ai

22. Art  The design for a mural is 16 in. wide and 9 in. high. What are the dimensions

of the largest possible complete mural that can be painted on a wall 24 ft wide by

14 ft high?

STEM 23. Architecture  You want to make a scale drawing of New York City’s Empire See Problem 5.

State Building using the scale 1 in. = 250 ft. If the building is 1250 ft tall,
how tall should you make the building in your scale drawing?

24. Cartography  A cartographer is making a map of Pennsylvania. She uses the scale

1 in. = 10 mi. The actual distance between Harrisburg and Philadelphia is about
95 mi. How far apart should she place the two cities on the map?

B Apply In the diagram below, △DFG ∼ △HKM. Find each of the following. G

25. the scale factor of △HKMto △DFG 26. m∠K 70Њ 27.5 M
29. GD 15
27. MGDH 28. MK 59Њ F H 18 K
D 30

30. Flags  A company produces a standard-size U.S. flag that is 3 ft

by 5 ft. The company also produces a giant-size flag that is

similar to the standard-size flag. If the shorter side of the

giant-size flag is 36 ft, what is the length of its longer side? hsm11gmse_0702_t05646.ai

31. a. Coordinate Geometry  What are the measures of ∠A, ∠ABC, ∠BCD, 6y F
∠CDA, ∠E, ∠F , and ∠G? Explain.

b. What are the lengths of AB, BC, CD, DA, AE, EF , FG, and AG? EC
c. Is ABCD similar to AEFG? Justify your answer. 2B

G

O 1A Dx
46

Lesson 7-2  Similar Polygons hsm11gmse_07024_4t150553.a

10

32. Think About a Plan  The Davis family is planning to

drive from San Antonio to Houston. About how far will Austin Houston
they have to drive?
Del Rio
• How can you find the distance between the two cities San Antonio Galveston
on the map?

• What proportion can you set up to solve the problem?

33. Reasoning  Two polygons have corresponding side Laredo Corpus
lengths that are proportional. Can you conclude that Scale Christi
the polygons are similar? Justify your reasoning. 1 cm : 112 km Brownsville

34. Writing  Explain why two congruent figures must also be
similar. Include scale factor in your explanation.

35. △JLK and △RTS are similar. The scale factor of △JLK to
△RTS is 3 : 1. What is the scale factor of △RTS to △JLK ?

36. Open-Ended  Draw and label two different similar quadrilaterals. Write a similarity
statement for each and give the scale factor.

Algebra  Find the value of x. Give the scale factor of the polygons.

37. △WLJ ∼ △QBV 38. GKNM ∼ VRPT
W x ϩ 6
J G 8.4 K P
x B5 4 xϩ4 3
L
Q8 V M 3x Ϫ 2 R T
4 6.3 V3

N

Sports  Choose a scale and make a scale drawing of each rectangular playing surface.

39. Ahssmoc1ce1rgfmielsdei_s 011700y2d_bt0y56604yd7.. ai 40. A volleyball court is 60 ft by 30 ft.

41. A tennis court is 78 ft by 36 ft. 42. Ahsfomot1b1agllmfiesled_is0376002f_t bt0y5166048ft..ai

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

43. Any two regular pentagons are similar. 44. A hexagon and a triangle are similar.

45. A square and a rhombus are similar. 46. Two similar rectangles are congruent.

STEM 47. Architecture  The scale drawing at the right is part of a floor plan for a home. Dining Kitchen
The scale is 1 cm = 10 ft. What are the actual dimensions of the family room? Family
room
C Challenge 48. The lengths of the sides of a triangle are in the extended ratio 2 : 3 : 4. The
perimeter of the triangle is 54 in. Master
a. The length of the shortest side of a similar triangle is 16 in. What are the lengths bedroom

of the other two sides of this triangle?
b. Compare the ratio of the perimeters of the two triangles to their scale factor.

What do you notice?

446 Chapter 7  Similarity

49. In rectangle BCEG, BC : CE = 2 : 3. In rectangle LJAW, LJ : JA = 2 : 3. Show that
BCEG ∼ LJAW.

50. Prove the following statement: If △ABC ∼ △DEF and △DEF ∼ △GHK , then
△ABC ∼ △GHK.

PERFORMANCE TASK MATHEMATICAL

Apply What You’ve Learned PRACTICES
MP 3
Look back at the information on page 431 about the graph Lillian wants to
adjust. The screens on pages 431 are shown again below. In the Apply What
You’ve Learned in Lesson 7-1, you determined the ratio of the width to the
height of Lillian’s calculator screen.

Plot1 Plot2 Plot3 WINDOW
\Y1 = 2X+3 Xmin = –10
\Y2 = –(1/2)X+1 Xmax = 10
\Y3 = Xscl = 1
\Y4 = Ymin = –10
\Y5 = Ymax = 10
\Y6 = Yscl =1
\Y7 =   =1  
TXres

a. Consider the “viewing rectangle” determined by the values of Xmin, Xmax, Ymin,
and Ymax. While the width and height of the calculator screen do not change, the
width and height of the viewing rectangle depend on the values entered for Xmin,
Xmax, Ymin, and Ymax. What are the width and height of the viewing rectangle for
Lillian’s graph shown above?

b. Is the viewing rectangle similar to Lillian’s rectangular calculator screen? Explain.

c. If the intersecting lines are to be graphed without distortion, what must be true
about the viewing rectangle?

d. If Lillian uses Xmin = -30 and Xmax = 30, will she be able to graph the lines
without distortion? Use similarity to explain why or why not.

Lesson 7-2  Similar Polygons 447

Concept Byte Fractals CMoamthmemonatCicosreFloStraidtea SSttaannddaarrddss

Use With Lesson 7-2 Extends MG-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.iteria
fsoimr tilraiarintyglcersitteorisaoflovretpriraonbglelemsstoansdoltvoe pprroovbelems
extension raenldattionpsrhoivpes rinelagteiomnsehtirpics fiingugreeosm. etric figures.

MP 7

Fractals are objects that have three important properties:

• You can form fractals by repeating steps. This process is called iteration.

• They require infinitely many iterations. In practice, you can continue until
the objects become too small to draw. Even then the steps could continue in
your mind.

• At each stage, a portion of the object is a reduced copy of the entire object at the
previous stage. This property is called self-similarity.

Example 1 Stage 04
1 unit
The segment at the right of length 1 unit is Stage 0 of a fractal tree. Draw Stage 1
and Stage 2 of the tree. For each stage, draw two branches from the top third of hsm11gmse_07fb_t05575.ai
each segment.

• sTeogdmraewntS. Ftargoem1,thfiinsdpothinetp, doirnatwthtwatoisse31gumneitnftrsoomf ltehnegttohp31oufnthite.

• To draw Stage 2, find the point that is 1 unit from the top of
9

each branch of Stage 1. From each of these points, draw two Stage 1 Stage 2

segments of length 1 unit. The length of each new branch is 1
9 3
1 1
of 3 unit which is 9 unit.

Amazingly, some fractals are used to describe natural formations suchhassmm1ou1ngtmainse_07fb_t05576.ai

ranges and clouds. In 1904, Swedish mathematician Helge von Koch made the Koch
Curve, a fractal that is used to model coastlines.

Example 2

The segment at the right of length 1 unit is Stage 0 of a Koch Curve. Draw Stage 0 1 unit
Stages 1–4 of the curve. For each stage, replace the middle third of each
segment with two segments, both equal in length to the middle third.

• For Stage 1, replace the middle third with two Stage 1 Stage 2
1 Stage 3
segments that are each 3 unit long. hsm11gmse_07fb_t05577.ai

• For Stage 2, replace the middle third of each Stage 4

segment of Stage 1 with two segments that are each
1
9 unit long.

• Continue with a third and fourth iteration.

448 Concept Byte  Fractals hsm11gmse_07fb_t05579.ai

Example 3 1 unit
Stage 0
The equilateral triangle at the right is Stage 0 of a Koch Snowflake. Draw Stage 1
of the snowflake by first drawing an equilateral triangle on the middle third of
each side. Then erase the middle third of each side of the original triangle.

• To draw an equilateral triangle on the middle third of a side, find the two
alereng13thun13itufnroitm. Eaanchensedgpmoiennttomf tuhset
points that side. From each point, draw a of
segment of make a 60° angle with the side

the original triangle.

Stage 1

Exercises hsm11gmse_07fb_t05581.ai

1. Draw Stage 3 of the fractal tree in Example 1. Stage 0 1 2
Length 1 ■ ■
Use the Koch Curve in Example 2 for Exercises 2–4.
2. Complete the table to find the length of the Koch Curve at each stage. hsm11gmse_07fb_t05582.ai
3. Examine the results of Exercise 2 and look for a pattern. Use this pattern to

predict the length of the Koch Curve at Stage 3 and at Stage 4.
4. Suppose you are able to complete a Koch Curve to Stage n.
a. Write an expression for the length of the curve.
b. What happens to the length of the curve as n increases?

5. Draw Stage 2 of the Koch Snowflake in Example 3.

Stage 3 of the Koch Snowflake is shown at the right. Use it and the earlier stages to
answer Exercises 6–8.

6. At each stage, is the snowflake equilateral?

7. a. Complete the table to find the perimeter at each stage.
Stage Number of Sides Length of a Side Perimeter

03 13
1■
2 48 1 ■ Stage 3
3
hsm11gmse_07fb_t05586.ai
■■

3■ ■■

b. Predict the perimeter at Stage 4.
c. Will there be a stage at which the perimeter is greater than 100 units? Explain.

8. SEnxeohrwscmifslae1kse14.gIasmntdhsi7es_stur0ug7egfebasb_totthu0at5tt5hth8ee4ar.reaeiiasonfothbeouKnodchoSnntohwe fplaekriem? eEtxeprloafinth. e Koch

Concept Byte  Fractals 449

7-3 Proving Triangles MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Similar
MG-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.isteimriailafroitrytrciraintegrlieasftoor
tsroialvnegplersobtolesmoslvaenpdrotoblpermosveanredlattoiopnrsohviepsreilnatgioeonmsheiptrsicin
gfiegoumrees.trAiclsfioguGre-Gs.PAEl.sBo.5MAFS.912.G-GPE.2.5

MP 1, MP 3, MP 4

Objectives To use the AA ∼ Postulate and the SAS ∼ and SSS ∼ Theorems
To use similarity to find indirect measurements

Are the triangles similar? How do you know? (Hint: Use a centimeter
ruler to measure the sides of each triangle.)

You’ve already
learned how to
decide whether
two polygons are
similar. This is a
special case of
that problem.

MATHEMATICAL

PRACTICES In the Solve It, you determined whether the two triangles are similar. That is, you
needed information about all three pairs of angles and all three pairs of sides.
In this lesson, you’ll learn an easier way to determine whether two triangles
are similar.

Lesson Essential Understanding  You can show that two triangles are similar
when you know the relationships between only two or three pairs of
Vocabulary corresponding parts.
• indirect

measurement

Postulate 7-1  Angle-Angle Similarity (AA =) Postulate

Postulate If . . . Then . . .
If two angles of one ∠S ≅ ∠M and ∠R ≅ ∠L △SRT ∼ △MLP
triangle are congruent
to two angles of another R L
triangle, then the
triangles are similar. S TM P

hsm11gmse_0703_t05714

450 Chapter 7  Similarity

What do you need Problem 1 Using the AA = Postulate W V
to show that the
triangles are similar? Are the two triangles similar? How do you know? 45Њ 45Њ
To use the AA ∙ R S
Postulate, you need to A △RSW and △VSB
B
prove that two pairs of ∠R ≅ ∠V because both angles measure 45°.
∠RSW ≅ ∠VSB because vertical angles are congruent.
angles are congruent. So, △RSW ∙ △VSB by the AA ∙ Postulate.

B △JKL and △PQR

∠L ≅ ∠R because both angles measure 70°. hKsm11gmse_Q0703_t05715

By the Triangle Angle-Sum Theorem, 70Њ 85Њ

m∠K = 180 - 30 - 70 = 80 and J 30Њ 70Њ P
m∠P = 180 - 85 - 70 = 25. Only one pair of LR

angles is congruent. So, △JKL and △PQR are not similar.

Got It? 1. Are the two triangles similar? How do you know?
a. 39Њ bh. sm11gm68sЊ e_0703_62t0Њ 5716

51Њ

Here are two other ways to determine whether two triangles are similar.

hsm11gmse_0703_t05717 hsm11gmse_0703_t05718

Theorem 7-1  Side-Angle-Side Similarity (SAS =) Theorem

Theorem If . . . Then . . .
If an angle of one triangle is △ABC ∙ △QRS
congruent to an angle of a AB = AC and ∠A ≅ ∠Q
second triangle, and the sides QR QS
that include the two angles
are proportional, then the AQ
triangles are similar.
BC S
R
You will prove Theorem 7-1 in Exercise 35.

Theorem 7-2  Side-Side-Side Similarity (SSS =) Theorem

Theorem Ihfs.m. .11gmse_0703_t05719 Then . . .

If the corresponding AB = AC = BC △ABC ∙ △QRS
sides of two triangles are QR QS RS
proportional, then the
A Q

triangles are similar.

BC S
R
You will prove Theorem 7-2 in Exercise 36.

Leshssomn 71-13 gmPrsoevi_n0g7T0ri3a_ngt0le5s7S2im0ilar 451

Proof Proof of Theorem 7-1: Side-Angle-Side Similarity Theorem A Q

Given: AB = AQCS , ∠A ≅ ∠Q
QR

Prove: △ABC ∙ △QRS BC

P<XlYa>n}foRrSP. Srohoofw: Cthhaoto△seQXXoYn∙R△QQsoRSthbayt QX = AB. Draw R S
the AA ∙ Postulate. A Q

Then use the proportion QX = QY and the given proportion
QR QS
AB AC
QR = QS  to show that AC = QY. Then prove that △ABC ≅ △QXY. hB sm11gmCseX_0703_t0Y5721

Finally, prove that △ABC ∙ △QRS by the AA ∙ Postulate. RS

Problem 2 Verifying Triangle Similarity hsm11gmse_0703_t05722

Are the triangles similar? If so, write a similarity statement for the triangles.

A S V 12 W
10 6 9 15
T X
U8

Use the side lengths to identify corresponding sides. 10 S V 12 W
Then set up ratios for each pair of corresponding sides. U8 6 9 15
T X
hsSmh1or1tgesmt ssidee_s0703_t0572XS4TV = 6 = 2
9 3
US 10 2
Longest sides WX = 15 = 3

Remaining sides TU = 8 = 2
VW 12 3

All three ratios are equal, so corresponding sides are proportional.
hsm11gmse_0703_t05723
△STU ∙ △XVW by the SSS ∙ Theorem.

How can you make B L 2 M
it easier to identify 8
corresponding sides
and angles? K 12 P 3 N

Sketch and label two

separate triangles.

∠K ≅ ∠K by the Reflexive Property of L M
Congruence. 8
8 ϩ 2 ϭ 10
hKKMsLm=11180g=m45saen_d07KK0NP3=_t110255=7542.5
K 12 P K 12 ϩ 3 ϭ 15 N
So, △KLP ∙ △KMN by the
SAS ∙ Theorem. ∠K is the included angle between
two known sides in each triangle.

452 Chapter 7  Similarity hsm11gmse_0703_t05726

Got It? 2. Are the triangles similar? If so, write a similarity statement for the triangles

and explain how you know the triangles are similar.

a. A 9 B G b. A 8 L 8 C
88 6
66 E 12 F W
C
6

E

Proof Problem 3hsmPr1o1vginmgseT_ri0a7n0g3l_ets0S5i7m2i7lar hGsm11gmse_070K3_t05728

Given: FG ≅ GH, F JL
JK ≅ KL, H
∠F ≅ ∠J

Prove: △FGH ∙ △JKL

The triangles are isosceles, You need to show hsmFi1nd1tgwmo psaeir_s 0of7c0or3re_spto0n5d7in2g 9
so the base angles are that the triangles
congruent. are similar. congruent angles and use the
AA ∙ Postulate to prove the
Statements triangles are similar.

1) FG ≅ GH, JK ≅ KL Reasons
2) △FGH is isosceles.
1) Given
△JKL is isosceles. 2) Def. of an isosceles △
3) ∠F ≅ ∠H, ∠J ≅ ∠L
4) ∠F ≅ ∠J 3) Base ⦞ of an isosceles △ are ≅.
5) ∠H ≅ ∠J
6) ∠H ≅ ∠L 4) Given
7) △FGH ∙ △JKL 5) Transitive Property of ≅
6) Transitive Property of ≅
7) AA ∙ Postulate

Got It? 3. a. Given:  MP } AC C P
Prove:  △ABC ∙ △PBM M

b. Reasoning  For the figure at the B
right, suppose you are given only that A
CA CB
PM = MB . Could you prove that the

triangles are similar? Explain.

hsm11gmse_0703_t05730

Lesson 7-3  Proving Triangles Similar 453

Essential Understanding  Sometimes you can use similar triangles to find
lengths that cannot be measured easily using a ruler or other measuring device.

You can use indirect measurement to find lengths that are difficult to measure directly.
One method of indirect measurement uses the fact that light reflects off a mirror at the
same angle at which it hits the mirror.

Problem 4 Finding Lengths in Similar Triangles

Rock Climbing  Before rock climbing, Darius wants to know how high he will climb.

He places a mirror on the ground and walks backward until he can see the top of

the cliff in the mirror. What is the height of the cliff? J

5.5 ft H x ft
T 6 ft V S
34 ft

Before solving for △HTV ∙ △JSV AA ∙ Postulate
x, verify that the
triangles are similar. HT = TV Corresponding sides of ∙ triangles are proportional.
△HTV ∙ △ JSV by JS SV
the AA∙ Postulate
because ∠T ≅ ∠S and 5.5 = 364 Substitute.
∠HVT ≅ ∠JVS. x

187 = 6x Cross Products Property

31.2 ≈ x Solve for x.

The cliff is about 31 ft high.

Got It? 4. Reasoning  Why is it important that the ground be flat to use the method of
indirect measurement illustrated in Problem 4? Explain.

454 Chapter 7  Similarity

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL

Are the triangles similar? If yes, write a similarity PRACTICES
statement and explain how you know they are similar.
4. Vocabulary  How could you use indirect

measurement to find the height of the flagpole at

1. R your school?

A E 35Њ 45Њ 5. Error Analysis  Which 9
100Њ B solution for the value of 4
x in the figure at the right 8x
Z is not correct? Explain.

2. A F 66

24 6 3 A. B.

hsmB 11g3 mseC_D0703_4t.05 5731E 4 = 8 8 = 4
8 x x 6

3. B 4x = 72 h4s8m=141xgmse_0703_t05734
U 16 G 12 = x
x = 18

hsm117g0mЊ 1s2e_0703_t057703Њ125 6. a. Compare and Contrast  How are the SAS
AF 20 E
Similarity Theorem and the SAS Congruence
hsm11gmse_0703_t05733
Phossmtu1la1tegmaliskee?_0H7o0w3a_rte0th5he7ys3md5if1fe1rgenmt?se_0703_t05736

b. How are the SSS Similarity Theorem and the

SSS Congruence Postulate alike? How are they

different?

Practice and Problem-Solving Exercises MATHEMATICAL

PRACTICES

A Practice Determine whether the triangles are similar. If so, write a similarity See Problems 1 and 2.
statement and name the postulate or theorem you used. If not, explain.

7. F 8. D8A 9. R 12 P 24 S
J M 6
6 16
H P 10 R N Q
GK T8

10. h3J2sKm1146g50mPse_L40300703RQ2_2t05737.1a1i . Shsm2U15Њ1gms35e3Њ5_NЊ0710103Њ_t05R7381.a2i. A 24 B

YA hs1m8 11gmse2_20970G3_t1015739.ai

C H 12 K

hsm11gmse_0703_t05740.ai hsmLes1s1ognm7-s3e  _P0ro7v0i3ng_tT0ri5a7ng4l1e.saSiimhilsamr 11gmse_0703_t057424.5a5i

13. Given:  ∠ABC ≅ ∠ACD 14. Given:  PR = 2NP, See Problem 3.
Proo f Prove:  △ABC ∙ △ACD Proof
PQ = 2MP

C Prove: △MNP ∙ △QRP
R

M

A DB P
N

Q

Indirect Measurement  Explain why the triangles are similar. Then find the See Problem 4.

distanhcsemre1p1rgemsesnete_d0b7y0x3._t05743.ai 16. hsm11gmse_0703_t05744.ai

15.

x x

90 ft 5 ft 6 in.

120 ft 135 ft

4 ft 10 ft
Mirror

17. Washington Monument  At a certain time of day, a 1.8-m-tall person standing
next to the Washington Monument casts a 0.7-m shadow. At the same time,
the Washington Monument casts a 65.8-m shadow. How tall is the Washington
Monument?

B Apply Can you conclude that the triangles are similar? If so, state the postulate or
theorem you used and write a similarity statement. If not, explain.

18. A F 18 E 24 19. L 20. P
S 16
32 48 38 D N N 12 K 12 D
M T 9
G
B 24 C

21. a. Are two isosceles triangles always similar? Explain.
b. Are two right isosceles triangles always similar? Explain.

22. Thisnmk 1A1bgoumtsaeP_l0a7n0  O3n_ta0s5u7n4n5y.daaiy,hascmlas1s1mgamtesuese_s0i7n0di3re_ctt0m57ea4s6u.raeimhenstmto1f1ingdmse_0703_t05747.ai

the height of a building. The building’s shadow is 12 ft long and your classmate’s

shadow is 4 ft long. If your classmate is 5 ft tall, what is the height of the building?
• Can you draw and label a diagram to represent the situation?
• What proportion can you use to solve the problem?

23. Indirect Measurement  A 2-ft vertical post casts a 16-in. shadow at the same time a
nearby cell phone tower casts a 120-ft shadow. How tall is the cell phone tower?

456 Chapter 7  Similarity

Algebra  For each pair of similar triangles, find the value of x.

24. 25. 2x Ϫ 4 26.
2x 10

x 39 x xϩ2
8 95 xϩ6
x ϩ 14
24

# # Pr2o7o .f PGhrisovmevne1::  1△PgQVmK#sReQ_is0Tis7,oS0sT3ce_#lte0Ts.5Q 7,4 PS8QT.a=i

P S
TQhVRs m11gmsPer2o_8o0. f 7GP0riov3ev_net::0   5AAB7BB4}9C.CaGDi =, BhCCsDm}1DA1GgCmse_0703_t05750.ai

KD

Q V RT A CG

29. Reasoning  Does any line that intersects two sides of a triangle and is parallel to the
third side of the triangle form two similar triangles? Justify your reasoning.

30. Chosnmst1r1ugctmiosnes _D07ra0w3a_nt0y 5△7A5B2C.awi ith m∠C = 30. Use ahstmra1ig1hgtemdgsee_an0d70co3m_tp0a5ss751.ai
to construct △LKJ so that △LKJ ∙ △ABC.
P WR
31. Reasoning  In the diagram at the right, △PMN ∙ △SRW . MQ QT
and RT are altitudes. The scale factor of △PMN to △SRW is 4 : 3. S

What is the ratio of MQ to RT ? Explain how you know. MN

32. Coordinate Geometry  △ABC has vertices A(0, 0), B(2, 4), and

Proof C(4, 2). △RST has vertices R(0, 3), S( - 1, 5), and T( - 2, 4). Prove that
△ABC ∙ △RST . (Hint: Graph △ABC and △RST in the coordinate plane.)

33. Write a proof of the following: Any two nonvertical parallel hsm11gmsey_0ᐍ7103_t0575E3.ai
Proof lines have equal slopes.
B ᐍ2
Given:  Nonvertical lines /1 and /2, /1 } /2,
EF and BC are # to the x-axis O x
A CD F
Prove:  ABCC = DEFF

34. Use the diagram in Exercise 33. Prove: Any two nonvertical lines with equal slopes
Proof are parallel.

C Challenge 35. Prove the Side-Angle-Side Similarity Theorem (Theorem 7-1). hsm11Agmse_0703_Qt05754.ai

Proo f Given:  QABR = AQCS, ∠A ≅ ∠Q BC
Prove:  △ABC ∙ △QRS R

S

hsm11gmse_0703_t05755.ai

Lesson 7-3  Proving Triangles Similar 457

36. Prove the Side-Side-Side Similarity Theorem (Theorem 7-2). B A Q
Proo f Given:  QABR = AQCS = BRCS S
Prove:  △ABC ∙ △QRS C
R

Standardized Test Prep

SAT/ACT 37. Complete the statement △ABC ∙ ? . By which postulate or hsm11gmsBe2_0N703_t05756.ai

theorem are the triangles similar? 8

△AKN ; SSS ∙ △ANK ; SAS ∙ A 12 C 3 K

△AKN ; SAS ∙ △ANK ; AA ∙

38. ∠1 and ∠2 are alternate interior angles formed by two parallel
lines and a transversal. If m∠2 = 68, what is m∠1?

22 68 112 122
hsm11gmse_0703_t05757.ai

39. The length of a rectangle is twice its width. If the perimeter of the rectangle is 72 in.,

what is the length of the rectangle?

12 in. 18 in. 24 in. 36 in.

Extended 40. Graph A(2, 4), B(4, 6), C(6, 4), and D(4, 2). What type of polygon is ABCD? Justify
Response your answer.

Mixed Review

TRAP = EZYD. Use the diagram at the right T 4R Y 45Њ D See Lesson 7-2.
to find the following. 4 A Z6 E See Lesson 1-4.

41. the scale factor of TRAP to EZYD P8

42. m∠R 43. DY 44. DE
PT

Use a protractor to find the measure of each angle. Classify the angle as acute,

right, obtuse, or straight. hsm11gmse_0703_t05758.ai

45. 46. 47. 48.

G et Ready!  To prepare for Lesson 7- 4, do Exercises 49 –52.

Algebhrsam  1Id1egnmtisfye_th0e70m3e_at0n5s7a5n9d.aeixtremes of each proporhtisomn.1T1hgemnsseo_lv0e70fo3r_xt0. 5761.ai See Lesson 7-1.

49. 8x = 1284 h5s0m. 1m121g=m1280s e_0703_t057605.1a.i x 15 2 = 9 52. xxhs-+m341=1g95mse_0703_t05762.ai
+ x

458 Chapter 7  Similarity

7 Mid-Chapter Quiz athXM

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

FO

Do you know HOW? Are the triangles similar? If so, write a similarity
statement and name the postulate or theorem you
1. A bookcase is 4 ft tall. A model of the bookcase is used. If not, explain.

6 in. tall. What is the ratio of the height of the model 11. L M 12. W 8 X 3 Y
bookcase to the height of the real bookcase? O
2Z
2. If a = 190, complete this statement: a = ■■. 9
b 9 NV

3. Are the two polygons shown below similar? If so, give
the similarity ratio of the first polygon to the second.

If not, explain.

D C H G 13. D 14. R O
8 6
hAsm21B14gms8e_07mC q_t05614.ahLism11gmT sFe_07mq_t05615.ai
A4B E 10 F

Solve each proportion. Algebra  Explain why the triangles are similar. Then
find the value of x.
4. 6y
= 18 5. 5 = x - 2 15. Hhs2m0 1K1gms2e8_I0x7Lm1J4q_t0561163. .haAsiEm6511gBms1x0e_07CmD q_t05616.ai
54 7 4

hsm11gmse_07mq_t05610.ai
6. On the scale drawing of a floor plan 2 in. = 5 ft. A

room is 7 in. long on the scale drawing. Find the

actual length of the room.

kABC M kDBF. Complete each statement. 17. In a garden, a birdbath 2 ft 6 in. tall casts an 18-in.
B
shhsamdo1w1gamt tshee_s0a7mmeqt_imt0e5a6n17o.aaki tree casts a 90-ft
DF shadow. How tall is the oak tree?

hsm11gmse_07mq_t05618.ai

A C Do you UNDERSTAND?
7. m∠A = m∠ ?  18. Writing  You find an old scale drawing of your home,
8. AB = BC
DB ■ but the scale has faded and you cannot read it. How
can you find the scale of the drawing?
9. A postcard is 6 in. by 4 in. A printing shop will
enhlasrmge11itgsmo sthea_t0t7hme lqo_ntg0e5r6s1id1e.aiis any length up to 19. Reasoning  The sides of one triangle are twice
3 ft. Find the dimensions of the biggest enlargement. as long as the corresponding sides of a second
triangle. What is the relationship between
10. Algebra  Find the value of x. the angles?

8 20. Error Analysis  Your classmate says that since all
10 x congruent polygons are similar, all similar polygons
must be congruent. Is he right? Explain.
24

hsm11gmse_07mq_t05612.ai Chapter 7  Mid-Chapter Quiz 459

7-4 Similarity in Right MCoamthmemonatCicosreFloStraidtea SSttaannddaarrddss
Triangles
MG-ASFRST.9B1.52 .GUs-SeR. T. .2s.i5m ilUasriety. .cr.isteimriailafroitrytrciraintegrlieasftoor
tsroialvnegplersobtolesmoslvaenpdrotoblpermosveanredlattoiopnrsohviepsreilnatgioeonmsheiptrsicin
gfiegoumrees.trAiclsfioguGre-Gs.PAEl.sBo.5MAFS.912.G-GPE.2.5

MP 1, MP 3, MP 4

Objective To find and use relationships in similar right triangles

Analyze the Draw a diagonal of a rectangular piece of paper 17 6
situation first. to form two right triangles. In one triangle, 2 5
Think about how draw the altitude from the right angle to the
you will match hypotenuse. Number the angles as shown. Cut 89
angles. out the three triangles. How can you match the 34
angles of the triangles to show that all three
triangles are similar? Explain how you know the
matching angles are congruent.

MATHEMATICAL

PRACTICES

In the Solve It, you looked at three similar right triangles. In this lesson, you will learn
new ways to think about the proportions that come from thesehssimmi1la1rgtmriasneg_l0e7s.0Y4o_ut05122.ai
began with three separate, nonoverlapping triangles in the Solve It. Now you will see
the two smaller right triangles fitting side-by-side to form the largest right triangle.

Essential Understanding  When you draw the altitude to the hypotenuse of a
right triangle, you form three pairs of similar right triangles.

T heorem 7-3

Lesson Theorem
The altitude to the hypotenuse of a right triangle divides the triangle into
Vocabulary two triangles that are similar to the original triangle and to each other.
• geometric mean

If . . . C Then . . .

△ABC is a right triangle △ABC ∼ △ACD
with right ∠ACB, and △ABC ∼ △CBD
△ACD ∼ △CBD
CD is the altitude to the A B
hypotenuse D D D

A CC B

460 Chapter 7  Similarity hsm11gmse_0704_t05124.ai

Proof Proof of Theorem 7-3 C B
AD
Given: Right △ABC with right ∠ACB
and altitude CD

Prove: △ACD ∼ △ABC, △CBD ∼ △ABC, △ACD ∼ △CBD

Statements Reasons hsm11gmse_0704_t05123.ai

1) ∠ACB is a right angle. 1) Given
2) Given
2) CD is an altitude. 3) Definition of altitude
4) Definition of #
3) CD # AB 5) All right ⦞ are ≅.
4) ∠ADC and ∠CDB are right angles.
5) ∠ADC ≅ ∠ACB, 6) Reflexive Property of ≅
7) AA ∼ Postulate
∠CDB ≅ ∠ACB
6) ∠A ≅ ∠A, ∠B ≅ ∠B 8) Corresponding ⦞ of ∼ △s are ≅.
7) △ACD ∼ △ABC, 9) All right ⦞ are ≅.
10) AA ∼ Postulate
△CBD ∼ △ABC
8) ∠ACD ≅ ∠B
9) ∠ADC ≅ ∠CDB
1 0) △ACD ∼ △CBD

Problem 1 Identifying Similar Triangles

What will help What similarity statement can you write relating the three WX
you see the triangles in the diagram?
corresponding ZY
vertices? YW is the altitude to the hypotenuse of right △XYZ, so you can use Y
Theorem 7-3. There are three similar triangles. X
Sketch the triangles
WX X hsm11WgmY se_070W4_t05125
separately in the same

orientation.

ZY Z YZ
△XYZ ∼ △YWZ ∼ △XWY

Ghsomt I1t?1g1m. as.e Wt_h0eh7ath0t rs4ei_metti0lrai5ar1int2ygl6setsaitnemtheendt icaagnraymou? write relating Q S R
P
b. Reasoning  From the similarity statement in

part (a), write two different proportions using

the ratio SR .
SP

hsm11gmse_0704_t05127

Lesson 7-4  Similarity in Right Triangles 461

Proportions in which the means are equal occur frequently in geometry. For any two

positive numbers a and b, the geometric mean of a and b is the positive number x such

that a = bx .
x

Problem 2 Finding the Geometric Mean
Multiple Choice  What is the geometric mean of 6 and 15?

H ow do you use 90 3 110 9 110 30
the definition of Definition of geometric mean
geometric mean? 6 = x
Set up a proportion x 15
with x in both means
positions. The numbers x2 = 90 Cross Products Property
6 and 15 go into the
extremes positions. x = 190 Take the positive square root of each side.

x = 3110 Write in simplest radical form.

The geometric mean of 6 and 15 is 3110. The correct answer is B.

Got It? 2. What is the geometric mean of 4 and 18?

In Got It 1 part (b), you used a pair of similar triangles to write a proportion with a
geometric mean.

Q short S long R ᭝SQP ϳ ᭝SPR

long short leg ϭ long leg
short short leg long leg
SP is the geometric mean
SQ ϭ SP of SQ and SR.
SP SR

P

This illustrates the first of two important corollaries of Theorem 7-3.

hsm11gms Ceo_r0o7l0la4r_yt0151t2o8 Theorem 7-3

Corollary If . . . Then . . .
The length of the altitude to the C
hypotenuse of a right triangle is the AD = CD
geometric mean of the lengths of the AD CD DB
segments of the hypotenuse.
B

Example  2 8 Sheypgomteenntussoefhsm1142 gϭms84e_0704_Ahytltp0ito5utd1een2ut9soe
4

You will prove Corollary 1 in Exercise 42.

hsm11gmse_0704_t05130

462 Chapter 7  Similarity

Corollary 2 to Theorem 7-3

Corollary If . . . Then . . .
The altitude to the hypotenuse of a right C
triangle separates the hypotenuse so that AB = AC
the length of each leg of the triangle is AD AC AD
the geometric mean of the length of the
hypotenuse and the length of the segment AB = CB
of the hypotenuse adjacent to the leg. CB DB

B

Example  2 Hypotenuhssem11g24mϭse12_0704SL_eegtgm05en1t3o1f hypotenuse

1 adjacent to leg
4

You will prove Corollary 2 in Exercise 43.

hsm11gmse_0704_t05132 s1 h
ᐍ1 s2
The corollaries to Theorem 7-3 give you ways to write proportions
using lengths in right triangles without thinking through a ᐍ2
the similar triangles. To help remember these corollaries,
consider the diagram and these properties.

Corollary 1 Corollary 2

s1 = a h = /s11, h = /2
a s2 /1 /2 s2
hsm11gmse_0704_t10552

Problem 3 Using the Corollaries 12
Algebra  What are the values of x and y? 4y

How do you decide x
which corollary
to use? Use Corollary 2. 4 ϩ 12 ϭ x Write a proportion. 4 ϭ y Use Corollary 1.
If you are using or x 4 y 12
finding an altitude,
use Corollary 1. If you x2 ϭ 64 Cross Products Property y 2 ϭ 48
are using or finding a Take the positihvessmqu1a1regrmoots.e_0y70ϭ4_͙t4085133
leg or hypotenuse, use x ϭ ͙64
Corollary 2.
xϭ8 Simplify. y ϭ 4 ͙3

Got It? 3. What are the values of x and y? 45
y

x

Lesson 7-4  Similharsitmy 1in1RgigmhsteT_ri0an7g0l4es_ t05134 463

Problem 4 Finding a Distance STEM B 20 in. A
D x
Robotics  You are preparing for a robotics
competition using the setup shown here.
Points A, B, and C are located so that
AB = 20 in., and AB # BC. Point D
is located on AC so that BD # AC
and DC = 9 in. You program the
robot to move from A to D and
to pick up the plastic bottle at
D. How far does the robot
travel from A to D?

C 9 in.

You can’t solve this x+ 9 = 20 Corollary 2
equation by taking 20 x
the square root. What
do you do? x2 + 9x = 400 Cross Products Property

Write the quadratic x2 + 9x - 400 = 0 Subtract 400 from each side.

equation in the (x - 16)(x + 25) = 0 Factor.

standard form  x - 16 = 0 or (x + 25) = 0 Zero-Product Property
ax2 + bx + c = 0.
Then solve by factoring x = 16 or x = - 25 Solve for x.

or use the quadratic Only the positive solution makes sense in this situation. The robot travels 16 in.

formula.

Got It? 4. From point D, the robot must turn right and move to point B to put the
bottle in the recycling bin. How far does the robot travel from D to B?

Lesson Check

Do you know HOW? Do you UNDERSTAND? MATHEMATICAL
Find the geometric mean of each pair of numbers.
PRACTICES

7. Vocabulary  Identify the following in △RST .

1. 4 and 9 2. 4 and 12 a. the hypotenuse RP
b. the segments of the hypotenuse

Use the figure to complete each proportion. c. the segment of the hypotenuse T

3. ge j adjacent to leg ST S
d
= e 4. = d def 8. Error Analysis  A classmate
■ ■ gh wrote an incorrect
j proportion to find x. Explain
5. ■f = ■f 6. j = ■ and correct the error. 3 hxsm13x1g=m8xse_0704_t05269
■ g 8

hsm11gmse_0704_t05268 hsm11gmse_0704_t05270

464 Chapter 7  Similarity