vocabulary - weeblymathbytonini.weebly.com/.../3/8/8/1/38811133/chapter_7.pdfvocabulary review...

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Vocabulary

Review

Chapter 7 182

7-1 Ratios and Proportions

1. Write a ratio to compare 9 red marbles to 16 blue marbles in three ways.

9 to 16

:

In simplest form, write the ratio of vowels to consonants in each word below.

2. comparison 3. geometry 4. ratio

to :

5. Cross out the ratio that is NOT equivalent to 12 to 8.

6 : 2 9 to 6 2416 48 : 32

Vocabulary Builder

proportion (noun) pruh PAWR shun

Other Word Form: proportional (adjective)

Definition: A proportion is an equation stating that two ratios are equal.

Examples: 23 5

812 and 12 5

510 are proportions.

Use Your Vocabulary

6. Write 3 or 6 to make each proportion true.

23 5 9

4

5

68 1

3 52

5

5

106

Underline the correct word to complete each sentence.

7. Distance on a map is proportion / proportional to the actual distance.

8. The number of ounces in 3 lb is in proportion / proportional to the number of ounces in 1 lb.

A proportion alwaysincludes an

equal sign, .

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Problem 1

Key Concept Properties of Proportions

width of bonsai to height of bonsai

Write as a fraction. Write using a colon.

:

Write using the

word “to.”

to

183 Lesson 7-1

Writing a Ratio

Got It? 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?

14. The bonsai is in. wide and in. tall.

15. Write the same ratio three different ways.

Cross Products Property In a proportion ab 5cd , where b 2 0 and d 2 0, the

product of the extremes a and d equals the product of the means b and c.

ab 5

cd

a ? d 5 b ?

5

Equivalent Forms of Proportions

Property 1 Property 2 Property 3

ab 5

cd is equivalent to a

b 5cd is equivalent to a

b 5cd is equivalent to

ba 5

dc . a

c 5bd .

a 1 bb

5c 1 d

d.

9. Identify the means and extremes in the proportion 23 54x .

Means and Extremes and

Identify the Property of Proportions each statement illustrates.

10. If 312 5

14 , then 31 5

124 .

11. If 45 58

10 , then 4(10) 5 5(8).

12. If 13 539 , then 31 5

93.

13. If 34 5xy , then 74 5

x 1 yy .

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Problem 3

the sum of theside lengths

Write

Relate the perimeteris

4x 60

Problem 4

7

9

perimeter

4x

Chapter 7 184

Using an Extended Ratio

Got It? 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?

16. Label the triangle at the right. Use the extended ratio to write an expression for each side length.

17. Complete the model to write an equation.

18. Use the justifications below to find the value of x.

4x 1 1 5 60 Write the equation.

? x 5 60 Combine like terms.

? x

5 60

Divide each side by .

x 5 Simplify.

19. Use the value of x to find each side length.

4x 5 4 ? 7 5 7 ? 5 ?

5 5 5

20. The lengths of the sides of the triangle are cm, cm, and cm.

Solving a Proportion

Got It? Algebra What is the solution of the proportion 92 5a

14?

21. Write a justification for each statement below.

92 5

a14

9(14) 5 2a

126 5 2a

1262 5

2a2

a 5 63

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Lesson Check

Math Success

Now Iget it!

Need toreview

0 2 4 6 8 10

Problem 5

28

73

4x

283 x

3x==

=

x6

y7 x

6

185 Lesson 7-1

Writing Equivalent Proportions

Got It? Use the proportion x6 5

y7. What ratio completes the equivalent

proportion 6x 5jj

? Justify your answer.

22. Use the diagram at the right. Draw arrows from the x and the 6 in the original proportion to the x and the 6 in the new proportion.

23. Circle the proportion equivalent to ab 5cd that you can use.

ba 5

dc a

c 5bd a 1 b

b 5c 1 d

d

24. Complete: x6 5y7 is equivalent to 6x 5

.

• Do you UNDERSTAND?

Error Analysis What is the error in the solution of the proportion at the right?

25. Circle the means of the proportion. Then underline the extremes.

3 4 7 x

26. Write each product.

Means ? 5 Extremes ? 5

27. What is the error in the solution of the proportion?

_______________________________________________________________________

28. Now solve the proportion correctly.

Check off the vocabulary words that you understand.

proportion means extremes Cross Products Property

Rate how well you can solve proportions.

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Vocabulary

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Similar Polygons7-2

Chapter 7 186

1. What does it mean when two segments are congruent?

_______________________________________________________________________

2. What does it mean when two angles are congruent?

_______________________________________________________________________

3. Measure each segment. Then circle the congruent segments.

Vocabulary Builder

similar (adjective) SIM uh lur

Other Word Forms: similarity (noun), similarly (adverb)

Definition: Things that are similar are alike, but not identical.

Math Usage: Figures that have the same shape but not necessarily the same size are similar.

Use Your Vocabulary

4. How are the two squares at the right similar?

______________________________________________________

5. How are the two squares NOT similar?

______________________________________________________

The symbol for similar is

.

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Problem 1

Problem 2

Key Concept Similar Polygons

CB

A D

IH

G J

K L

M Z Y

XW

N

10 20

1515

187 Lesson 7-2

Understanding Similarity

Got It? DEFG , HJKL. What are the pairs of congruent angles? What is the extended proportion for the ratios of the lengths of corresponding sides?

11. Complete each congruence statement. 12. Complete the extended proportion.

/D> / DEHJ 5

EF5

KL5

/E > /

/K> /

/L > /

A scale factor is the ratio of the lengths of corresponding sides of similar triangles.

Determining Similarity

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

13. Circle the short sides of each rectangle. Underline the long sides.

KL LM MN NK

WX XY YZ ZW

14. Write the ratios of corresponding sides in simplest form.

KLXY 5

1015 5 LM

YZ 515

5 MNZW 5

155 NK

WX 5 5

Two polygons are similar polygons if corresponding angles are congruent and if the lengths of corresponding sides are proportional.

ABCD , GHIJ . Draw a line from each angle in Column A to its corresponding angle in Column B.

Column A Column B

6. /A /H

7. /B /J

8. /C /G

9. /D /I

10. Complete the extended proportion to show that corresponding sides of ABCD and GHIJ are proportional.

ABGH 5

BC5

IJ5

AD

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d.Problem 4

Problem 3

D

E

A B

7.56

5

x

y F

G C

9

Chapter 7 188

15. Place a ✓ in the box if the statement is correct. Place an ✗ if it is incorrect.

KLMN , XYZW and the scale factor is 23.

KLMN , XYZW and the scale factor is 34.

The polygons are not similar.

Using Similar Polygons

Got It? ABCD , EFGD. What is the value of y?

16. Circle the side of ABCD that corresponds to EF .

AB BC CD AD

17. Use the justifications at the right to find the value of y.

EF

5EDAD Corresponding sides of similar polygons are proportional.

y

569 Substitute.

9y 5 Cross Products Property

y 5 Divide each side by 9.

Using Similarity

Got It? A rectangular poster’s design is 6 in. high by 10 in. wide. What are the dimensions of the largest complete poster that will fit in a space 3 ft high by 4 ft wide?

18. Determine how many times the design can be enlarged.

Height: 3 ft 5 in. Width: 4 ft 5 in.

in. 4 6 in. 5 6 in. 4 10 in. 5 4.8

The design can be enlarged at most times.

19. Let x represent the height of the poster. Write a proportion and solve for x.

20. The largest complete poster that will fit is in. by in.

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Lesson Check

Now Iget it!

Need toreview

0 2 4 6 8 10

Math Success

Problem 5

B

A R

P

S

Scale 1 cm : 200 m

cm

189 Lesson 7-2


The triangles at the right are similar. What are three similarity statementsfor the triangles?

25. The triangles are n and n .

26. /A > / /B > / /S > /

27. nABS , nBSA , nSAB ,


similar extended proportion scale factor scale drawing

Rate how well you can identify and apply similar polygons.

Using a Scale Drawing

Got It? Use the scale drawing of the bridge. What is the actual height of the towers above the roadway?

21. Use a centimeter ruler to measure the height of the towers above the roadway in the scale drawing. Label the drawing with the height.

22. Identify the variable.

Let h 5 the 9 of the towers.

23. Use the information on the scale drawing to write a proportion. Then solve to find the value of the variable.

QHint: 1

2005

tower height in drawing (cm)

actual height (m)R

24. The actual height of the towers above the roadway is m.

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Vocabulary

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Chapter 7 190

7-3 Proving Triangles Similar

Write the converse of each theorem.

1. If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

If

,

then

.

2. If a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

If

,

then

.

Vocabulary Builder

verify (verb) VEHR uh fy

Related Word: proof (noun)

Definition: To verify something means to find the truth or accuracy of it.

Math Usage: A proof is a way to verify a conjecture or statement.

Use Your Vocabulary

Write T for true or F for false.

3. You can verify that two triangles are similar by showing that corresponding

angles are proportional.

4. You can use properties, postulates, and previously proven theorems to verify steps in a proof.

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d.Key Concept Postulate 7–1, Theorem 7–1, Theorem 7–2

Problem 1

Problem 2

A B

C

G

E F

9

6 6 88

12

39

51

191 Lesson 7-3

Verifying Triangle Similarity

Got It? Are the triangles similar? If so, write a similarity statement for the triangles and explain how you know the triangles are similar.

8. Write ratios for each pair of corresponding sides.

9. Circle the postulate or theorem you can use to verify that the triangles are similar.

AA , Postulate SAS , Theorem SSS , Theorem

10. Complete the similarity statement.

nABC ,n

Postulate 7-1 Angle-Angle Similarity (AA ,) Postulate If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

Theorem 7-1 Side-Angle-Side Similarity (SAS ,) Theorem If an angle of one triangle is congruent to an angle of a second triangle, and the sides that include the two angles are proportional, then the triangles are similar.

Theorem 7-2 Side-Side-Side Similarity (SSS ,) Theorem If the corresponding sides of two triangles are proportional, then the triangles are similar.

5. Write the postulate or theorem that proves the triangles similar.

63

42

Using the AA,Postulate

Got It? Are the two triangles similar? How do you know?

6. Complete the diagram.

7. Are the triangles similar? Explain.

_________________________________________________________

__________________________________________________________

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Problem 4

Problem 3

AA ~ Postulate Given

Vertical angles are congruent.

S

H

34 ft34 ft

x ft

5.5 ft5.5 ft

S

x ft

JJ

H

6 ft VT 6 ft VT

C

AB

M

P

Chapter 7 192

Proving Triangles Similar

Got It? Given: AC uu MP Prove: nABC , nPBM

11. The proof is shown below. Write a reason from the box for each statement.

Statements Reasons

1) AC uu MP 1)

2) /A > /P 2) If parallel lines are cut by a transversal, alternate interior angles are congruent.

3) /ABC > /PBM 3)

4) nABC , nPBM 4)

Finding Lengths in Similar Triangles

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

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.

12. If the ground is NOT flat, will /HTV and /JSV be right angles? Yes / No

13. If the ground is NOT flat, will you be able to find congruent angles? Yes / No

14. Why is it important that the ground be flat? Explain.

___________________________________________________________________

___________________________________________________________________

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Lesson Check • Do you UNDERSTAND?

Math Success

Shortest Side Longest Side Third SideTriangle

Larger

Smaller

6

Now Iget it!

Need toreview

0 2 4 6 8 10

x

94

8

6 6

193 Lesson 7-3

Error Analysis Which solution for the value of x in the figure at the right is not correct? Explain.

A.

x = 184x = 72

8x=4

8

B. 46=

12 = x

8x

48 = 4x

15. Write the side lengths of the triangles.

16. Write ratios to compare the lengths of the corresponding sides.

shortest sides: longest sides: third sides:

17. Cross out the proportion that does NOT show ratios of corresponding sides.

96 564 9

4 5x8 x

8 564 x

8 596

18. Cross out the solution that does NOT show ratios of corresponding sides.

Solution A Solution B

19. Explain why the solution you crossed out does NOT show the correct value of x.

_______________________________________________________________________


indirect measurement similar triangles

Rate how well you can prove triangles similar.

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Vocabulary

Review

AD

C

B

Chapter 7 194

7-4 Similarity in Right Triangles

Underline the correct word to complete the sentence.

1. The altitude of a triangle is a segment from a vertex to the opposite

side that is parallel / perpendicular to the opposite side.

2. In an isosceles triangle, the altitude to the base divides the triangle

into two congruent / isosceles triangles.

3. Circle the altitude of nABC .

AB AC BC CD

Vocabulary Builder

geometric mean (noun) jee uh MEH trik meen

Definition: For any two positive numbers a and b, the geometric mean of a and b is the positive number x such that ax 5

xb .

Example: The geometric mean of 4 and 10 is the value of x in 4x 5

x10, or

x 5 2"10.

Use Your Vocabulary

4. Multiple Choice Which proportion can you use to find the geometric mean of 5 and 15?

x5 5

x15 5

x 515x 5

x 5x

15 515 5

xx

Underline the correct equation to complete each sentence.

5. The geometric mean x of a and b is x 5 "ab / x 5 ab .

6. The geometric mean x of 3 and 7 is x 5 "21 / x 5 21 .

7. Circle the geometric mean of !3 and !3.

!3 3 3!3 !33

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d.Key Concept Theorem 7-3 and Corollaries 1 and 2

A D

C

B

A D

C

B

LN

MP

a b

cx y

m

A

A

D

C BC

DD

C

B

195 Lesson 7-4

Theorem 7-3 Th e altitude to the hypotenuse of a right triangle divides the triangle intotwo triangles that are similar to the original triangle and to each other.

Corollary 1 to Theorem 7-3

Th e length of the altitude to the hypotenuse of a right triangle is the geometric mean of the lengths of the segments of the hypotenuse.

If . . . Then . . .

ADCD 5

CDDB

Corollary 2 to Theorem 7-3

Th e altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the triangle is the geometric mean of the length of the hypotenuse and the length of the segment of the hypotenuse adjacent to the leg.

If . . . Then . . .

ABAC 5

ACAD

ABCB 5

CBDB

8. nLMN is a right triangle with right /LMN . NP is the altitude to the hypotenuse. Complete the similarity statements.

nLMN , n

nLMN , n

nLNP , n

Use the triangle at the right. Write Corollary 1 or Corollary 2 for each proportion.

9. ca 5

ax

10. xm 5

my

11. cb 5

by

If . . .

nABC is a right triangle with right /ACB, and

CD is the altitude to the hypotenuse

Then . . .

nABC , nACD

nABC , nCBD

nACD , nCBD

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Problem 1

Problem 2

Problem 3

RQ

P

S

yx

4 5

Chapter 7 196

Identifying Similar Triangles

Got It? What similarity statement can you write relating the three triangles in the diagram?

12. Write the names of the triangles.

nRPQ n n

13. Write the three right angles. 14. Write the three smallest angles.

/RPQ / / /QRP / /

15. Use your answers to Exercises 13 and 14 to write three similarity statements beginning with the vertex of the smallest angle in each triangle and ending with the vertex of the right angle.

nRQP , n nRQP , n n , n

Finding the Geometric Mean

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

16. Use the justifications below to find the geometric mean.

4x 5

x Definition of geometric mean

x2 5 Cross Products Property

x 5 Å Take the positive square root of each side.

x 5 Å Write in simplest radical form.

Using the Corollaries

Got It? What are the values of x and y?

Underline the correct word to complete each sentence.

17. x is the length of a leg of the largest 18. y is the length of the altitude of the largest

triangle, so use Corollary 1 / Corollary 2 triangle, so use Corollary 1 / Corollary 2

to find the value of x. to find the value of y.

19. The values of x and y are found below. Write a justification for each step.

4x 5

x4 1 5 4

y 5y5

x 2 5 36 y 2 5 20

x 5 "36 y 5 "20

x 5 6 y 5 2"5

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Now Iget it!

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0 2 4 6 8 10

Math Success

Lesson Check

Problem 4

x20 in.

9 in. 16 in.A

B

DC

S T

PR

197 Lesson 7-4


geometric mean altitude similarity

Rate how well you understand similar right triangles.


Vocabulary Identify the following in nRST .

23. The hypotenuse is .

24. The segments of the hypotenuse are and .

25. The segment of the hypotenuse adjacent to leg ST is .

Finding a Distance

Got It? Points A, B, and C are located so that AB 5 20 in., and AB ' BC . Point D is located on AC so that BD ' AC and DC 5 9 in. You program a robot to move from A to D and to pick up a plastic bottle at D. From point D, the robot must turn right and move to point B to put the bottle in a recycling bin. How far does the robot travel from D to B?

20. Place a ✓ in the box if the statement is correct. Place an ✗ if it is incorrect.

I know the length of the hypotenuse of nABC .

I know the lengths of the segments of the hypotenuse of nABC .

I know the length of the altitude of nABC .

I can use Corollary 1 to solve the problem.

21. Find the length of BD.

22. The robot travels in. from D to B.

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Vocabulary

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CB

AD

Proportions in Triangles 7-5

Chapter 7 198

1. Circle the model that can form a proportion with 1015.

2. Circle the ratios that you can use to form a proportion.

12 3

4 25100 75

100

3. Cross out the proportion that does NOT have the same solution as the others.

1217 5

n20 12

n 51720 n

17 52012 20

n 51712

Vocabulary Builder

bisector (noun) BY sek tur

Other Word Form: bisect (verb)

Definition: A bisector divides a whole into two equal parts.

Math Usage: A bisector is a point, segment, ray, or line that divides an angle or a segment into two congruent angles or segments.

Use Your Vocabulary

Use the diagram at the right. Complete each statement with the correct word from the list below. Use each word only once.

bisects bisector bisected

4. BD) is the 9 of /ABC .

5. /ABC is 9 by BD) .

6. BD) 9 /ABC .

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Problem 1

Theorem 7-4 Side-Splitter Theorem and Its Corollary

X

R S

Y

Q

A W

X

Y

a

b

c

B

C

a 4

a 12

18

199 Lesson 7-5

Using the Side-Splitter Theorem

Got It? What is the value of a in the diagram at the right?

12. The value of a is found below. Use one of the reasons in the box to justify each step.

Cross Products Property Divide each side by 6.Side-Splitter Theorem Simplify.Subtract 12a from each side.

aa 1 4 5

1218

18a 5 12a 1 48

18a 2 12a 5 12a 2 12a 1 48

6a 5 48

6a6 5

486

a 5 8

Side-Splitter TheoremIf a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

If *RS)u u*XY), then XR

RQ 5SQ

.

7. If XR 5 4, RQ 5 4, and YS 5 5, then SQ 5 .

8. If XR 5 3, RQ 5 6, and YS 5 4, then SQ 5 .

Corollary to the Side-Splitter TheoremIf three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

If a 6 b 6 c , then ABBC 5

WXXY .

Complete each proportion.

9. BCAB 5

XY 10. BA 5YX

XW 11. ACAB 5 WX

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d.

Problem 3

Problem 2

Site A Site B

8 yd

9 yd 7.2 yd

6.4 yd

Site C

A

BC D

y 24

9.6 16

hsm11_gemc_0705_t93531

Think Write

I can use the Triangle-Angle-Bisector

Theorem to write a proportion.

Then I can use the Cross-Products

Property.

9.616

y

16y

16

y

1616 y

Now I divide each side byand simplify.

Chapter 7 200

Finding a Length

Got It? Camping Three campsites are shown in the diagram. What is the length of Site C along the road?

13. Let y be the length of Site C along the road. Use the justifications at the right to find the value of y.

y

7.25

6.4

Corollary to Side-Splitter

Theorem

? y 5 46.08 Cross Products Property

? y

546.08

Divide each side by the

coefficient of y.

y 5 Simplify.

14. The length of Site C along the road is yd.

Using the Triangle-Angle-Bisector Theorem

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

15. Complete the reasoning model below.

16. The value of y is .

Triangle-Angle-Bisector Theorem

If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle.

If AD) bisects /CAB, then CD

DB 5CABA .

Theorem 7-5 Triangle-Angle-Bisector Theorem

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Lesson Check

Lesson Check

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Math Success

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201 Lesson 7-5

• Do you know HOW?


What is the value of x in the figure at the right?

17. Circle the proportion you can use to solve the problem.

1030 5

x45 x

10 53045 x

x 1 10 53045 10

x 1 10 53045

18. Solve the proportion.

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

19. Cross out the lengths that are NOT parts of the sides intersected by the parallel line.

2 2.4 3 7 x y

20. Can you use the Side-Splitter Theorem to find x? Yes / No

21. Can you use the Side-Splitter Theorem to find y? Yes / No

22. Explain what is wrong with your classmate’s statement.

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bisector proportion Side-Splitter Theorem

Rate how well you understand side and angle bisectors.

HSM11GEMC_0705.indd 201 3/5/09 8:43:39 AM

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