north valleys high school mathematics geometry 2...©gm2t0d1v2x3kustnaaosmoqf xtmwzagrmegdlblqcd.3k...

North Valleys High School Mathematics Geometry 2

Student Name: ________________________________ Period: _______

The Topics for this section: Solving Proportions based on Triangles

Dilations Similarity and Similar Triangles

Triangles Divided by Parallel Lines Midsegments of Triangles

Proving the Pythagorean Theorem

At the end of this section the student should:

Solve proportions arithmetically and algebraically Apply proportions to a variety of triangle solutions Understand the rules for triangle proportions and midsegments Know and apply the rules for similarity Draw dilations and do calculations based on dilations Understand the basis of the Pythagorean Theorem from a Completed Proof

Contents

Title Page

Solving Proportions 1

Practice 7-1 Ratios and Proportions 2

Dilation Practice 1 3

Dilation Practice 2 4

Skills Practice 7-2 Similar Polygons 6

Skills Practice/Practice 7-3 Similar Triangles 8

Word Problems 7-3 Similar Triangles 10

Similarity 2 11

Skills Practice/Practice 7-5 Parts of Similar Triangles 13

Word Problems 7-5 Parts of Similar Triangles 15

Solving Triangles with Proportional Parts 16

Triangles with Parallel Lines (Side Splitter) 1 19

Triangles with Parallel Lines (Side Splitter) 2 20

Midsegments 1 21

Midsegments 2 23

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North Valleys High School Geometry 1ID: 1

Name___________________________________

Period____Date________________©W s2W0d172A dKyuQtaaG hS6ouflt1waasrQeO kLSLLC4.m 7 tA4lIl2 IrNi9gGhctosg sr5epsVetr8vTeYdQ.vSolving ProportionsSolve each proportion.

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2)r

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11)m

m12)

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14)r

r

15)x x

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17)bb

18)n n

19)x x

20)nn

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Chapter 7 8 Glencoe Geometry

PracticeRatios and Proportions

1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fat are saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese.

2. FARMING The ratio of goats to sheep at a university research farm is 4:7. The number of sheep at the farm is 28. What is the number of goats?

3. QUALITY CONTROL A worker at an automobile assembly plant checks new cars for defects. Of the first 280 cars he checks, 4 have defects. If 10,500 cars will be checked this month, predict the total number of cars that will have defects.

Solve each proportion.

4. 5 − 8 = x −

12 5. x −

1.12 = 1 −

5 6. 6x −

27 = 43

7. x + 2 − 3 = 8 −

9 8. 3x - 5 −

4 = -5 −

7 9. x -2 −

4 = x + 4 −

2

10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet. Find the measure of each side of the triangle.

11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches. Find the measure of each side of the triangle.

12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters. Find the measure of each side of the triangle.

13. The ratio of the measures of the three angles is 4:5:6. Find the measure of each angle of the triangle.

14. The ratio of the measures of the three angles is 5:7:8. Find the measure of each angle of the triangle.

15. BRIDGES A construction worker is placing rivets in a new bridge. He uses 42 rivets to build the first 2 feet of the bridge. If the bridge is to be 2200 feet in length, predict the number of rivets that will be needed for the entire bridge.

7-1

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MATH-DRILLS.COM MATH-DRILLS.COM MATH-DRILLS.COM MATH-DRILLS.COM

Dilations (A) Instructions: Draw and label the dilated image for each triangle.

Label the center of dilation.

1) 2)

3) 4)

5) 6)

Page 3 of 24

Geometry Practice: G.G.58 #1www.jmap.org NAME:_____________________________

P.I. G.G.58: Define, investigate, justify, and applysimilarities (dilations and the composition ofdilations and isometries)

1. Graph EF with E(–2, –2) and F(3, –4). Thengraph its dilation with a scale factor of 2.

[1]

x

y

–10 10

–10

10

2. Graph GH with G(3, –3) and H(–2, 1). Thengraph its dilation with a scale factor of 2.

[2]

x

y

–10 10

–10

10

3. Graph IJ with I(–4, 4) and J(1, 3). Thengraph its dilation with a scale factor of 2.

[3]

x

y

–10 10

–10

10

4. Graph CD with C(4, –3) and D(1, 1). Thengraph its dilation with a scale factor of 2.

[4]

x

y

–10 10

–10

10

5. Graph JK with J(2, 3) and K(–4, 4). Thengraph its dilation with a scale factor of 2.

[5]

x

y

–10 10

–10

10

6. Graph GH with G(4, –4) and H(–2, 3). Thengraph its dilation with a scale factor of 2.

[6]

x

y

–10 10

–10

10

Page 4 of 24


7. Graph BC with B(3, –2) and C(–1, –3). Thengraph its dilation with a scale factor of 1.5.

[7]

x

y

–10 10

–10

10

8. Graph AB with A(–4, 2) and B(1, 4). Thengraph its dilation with a scale factor of 2.5.

[8]

x

y

–10 10

–10

10

9. Graph FG with F(3, –4) and G(2, –1). Thengraph its dilation with a scale factor of 0.5.

[9]

x

y

–10 10

–10

10

10. Graph DE with D(1, 1) and E(2, –2). Thengraph its dilation with a scale factor of 1.5.

[10]

x

y

–10 10

–10

10

11. Graph AB with A(4, 1) and B(3, 3). Thengraph its dilation with a scale factor of 2.5.

[11]

x

y

–10 10

–10

10

12. Graph EF with E(–2, –4) and F(–1, 2). Thengraph its dilation with a scale factor of 0.5.

[12]

x

y

–10 10

–10

10

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Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning.

1. 2.

7.5

7.5

7.57.5

Z Y

XW

3

3

33S R

QP

Each pair of polygons is similar. Find the value of x.

3. 4.

5. 6.

9

103

x + 1

5

P

S

RU V

WT

Q

10

8

x + 2

x - 1

SP

M N

L

T

x + 5

x + 5

9

3

44 S

W

XY

T

U26

14

12 13

B C

DA

13

76 x

G

E H

F

10.5

6 9

59° 35°A C

B

7

4 659° 35°D F

E

Skills PracticeSimilar Polygons

7-2

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PracticeSimilar Polygons

Determine whether each pair of figures is similar. If so, write the similarity statement and scale factor. If not, explain your reasoning. 1.

25 12

9

14.4

1524

20

15

JM Q R

SP

KL 2.

2412

14 162118

VUAC

B T

Each pair of polygons is similar. Find the value of x.

3.

14

10

x + 6

x + 9

C N MD

A B LP

4.

6 12

40°

40°

x + 1

x - 3AF

E

D

B C

5. PENTAGONS If ABCDE ∼ PQRST, find the scale factor of ABCDE to PQRST and the perimeter of each polygon.

6. SWIMMING POOLS The Minnitte family and the neighboring Gaudet family both have in-ground swimming pools. The Minnitte family pool, PQRS, measures 48 feet by 84 feet. The Gaudet family pool, WXYZ, measures 40 feet by 70 feet. Are the two pools similar? If so, write the similarity statement and scale factor.

7-2

2015

10

21

48 ft

84 ft

70 ft

40 ftP

Q

S W

Z

X

Y

R

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Determine whether each pair of triangles is similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning.

1. 2.

3. 4.

2170°

15

MJ

P

1470°10

U

S

T

30°60° RQ

M

S

K

T

ALGEBRA Identify the similar triangles. Then find each measure.

5. AC 6. JL

1512

D

E CB

Ax + 1

x + 5

16

4M

NL

J

K

x + 18

x - 3

7. EH 8. VT

12

9

6

9

D

E

FH

Gx + 5

6

14

S

U

V

R T

3x - 3

x + 2

12

128

9

9 6

C P

QR

A

B

Skills PracticeSimilar Triangles

7-3

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PracticeSimilar Triangles

Determine whether the triangles are similar. If so, write a similarity statement. If not, what would be sufficient to prove the triangles similar? Explain your reasoning.

1. 2.

ALGEBRA Identify the similar triangles. Then find each measure.

3. LM, QP 4. NL, ML

5. 6.

8

N

J LK

Mx + 5

6x + 2

24

12

18N

P

QL

M

x + 3x - 1

42°

42°

24

1812

16J

A K

Y S

W

7-3

10 5

128

86

x + 7 x - 1

x + 1

x + 33

4

7. INDIRECT MEASUREMENT A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches casts an 8-foot shadow.

a. Write a proportion that can be used to determine the height of the lighthouse.

b. What is the height of the lighthouse?

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Word Problem PracticeSimilar Triangles

A

B

D

E

C

1. CHAIRS A local furniture store sells two versions of the same chair: one for adults, and one for children. Find the value of x such that the chairs are similar.

18˝

16˝

X

12˝

2. BOATING The two sailboats shown are participating in a regatta. Find the value of x.

168.5 in. 220 in.

X in. 264 in.

40° 40°

3. GEOMETRY Georgia draws a regular pentagon and starts connecting its vertices to make a 5-pointed star. After drawing three of the lines in the star, she becomes curious about two triangles that appear in the figure, �ABC and �CEB. They look similar to her. Prove that this is the case.

4. SHADOWS A radio tower casts a shadow 8 feet long at the same time that a vertical yardstick casts a shadow half an inch long. How tall is the radio tower?

5. MOUNTAIN PEAKS Gavin and Brianna want to know how far a mountain peak is from their houses. They measure the angles between the line of site to the peak and to each other’s houses and carefully make the drawing shown.

Gavin

Brianna

Peak

2 in.

2.015 in.0.246 in.

90˚83˚

The actual distance between Gavin and Brianna’s houses is 1 1−

2 miles.

a. What is the actual distance of the mountain peak from Gavin’s house? Round your answer to the nearest tenth of a mile.

b. What is the actual distance of the mountain peak from Brianna’s house? Round your answer to the nearest tenth of a mile.

7-3

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P.I. G.G.45: Investigate, justify, and apply theorems about similar triangles

1. Complete the following when IJK ~ LMN .a) m K = m _____

b) KIJI

= NL?

2. In QRS , QR 7, RS 13, and m R 46. In UV T , V T 14, TU 28, and m T 46. Statewhether the triangles are similar, and if so, write a similarity statement.

3. Find the scale factor that maps ABC onto A B C if A B C( , ), ( , ), ( , ),2 0 0 4 6 0 A ( , ),3 0B ( , ),0 6 C ( , ).9 0 How are the figures related? Explain.

4. ABC with vertices A B( , ), ( , ),2 1 2 5 and C( , )2 4 is similar to MNO with vertices M ( , )3 1 andN ( , ).3 9 Find four possibilities for the coordinates of vertex O.

5. A lamppost is 6 feet high and casts an 8-foot shadow. At the same time of day, a flagpole directlybehind the lamppost casts a 28-foot shadow.

28 ft 8 ft

H

6 ft

Which proportion can be used to find the height, H, of the flagpole?

[A] H28

68

= [B] H8

628

= [C] 828 6

= H [D] H28

86

=

Page 11 of 24


6. At the same time of day, a man who is 52.8 inches tall casts a 68.8-inch shadow and his son casts a43-inch shadow. What is the height of the man’s son?

52.8 in.

68.8 in. 43 in.

[A] 33 in. [B] 85.8 in. [C] 111.8 in. [D] 34 in.

7. Two ladders are leaning against a wall at the same angle as shown. How far up the wall does the shorterladder reach?

30 ft48 ft

16 ft

[A] 8 ft [B] 10 ft [C] 6 ft [D] 20 ft

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7-5 Skills PracticeParts of Similar Triangles

Find x.

1. 2. 7

7

22

x 5.25

5.25

15

33

10

x

3. 4.

20

10x

22

18

1212.6

x

5. If �RST ∼ �EFG, −−−

SH is an 6. If �ABC ∼ �MNP, −−−

AD is an altitude of �RST,

−− FJ is an altitude of altitude of �ABC,

−−− MQ is an altitude of

�EFG, ST = 6, SH = 5, and FJ = 7, �MNP, AB = 24, AD = 14, andfind FG. MQ = 10.5, find MN.

JGE

F

HTR

S

C

A

BD P NQ

M

Find the value of each variable.

7. 8.

26

128

m

7

20

24

x

016_037_GEOCRMC07_890516.indd 32 6/17/08 11:16:37 PM

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Less

on

7-5


PracticeParts of Similar Triangles

ALGEBRA Find x.

1.

3230

24x

2.

2625

39x

3. 402x + 1 25x + 4

4.

5. If �JKL ∼ �NPR, −−−KM is an 6. If �STU ∼ �XYZ,

−−−UA is an

altitude of �JKL, −−PT is an altitude altitude of �STU,

−−ZB is an altitude

of �NPR, KL = 28, KM = 18, and of �XYZ, UT = 8.5, UA = 6, and PT = 15.75, find PR. ZB = 11.4, find ZY.

MJ L

K

TN R

P

YX B

Z

TS A

U

7. PHOTOGRAPHY Francine has a camera in which the distance from the lens to the film is 24 millimeters.

a. If Francine takes a full-length photograph of her friend from a distance of 3 meters and the height of her friend is 140 centimeters, what will be the height of the image on the film? (Hint: Convert to the same unit of measure.)

b. Suppose the height of the image on the film of her friend is 15 millimeters. If Francine took a full-length shot, what was the distance between the camera and her friend?

7-5

x28

20 30

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Word Problem PracticeParts of Similar Triangles

1. FLAGS An oceanliner is flying two similar triangular flags on a flag pole. The altitude of the larger flag is three times the altitude of the smaller flag. If the measure of a leg on the larger flag is 45 inches, find the measure of the corresponding leg on the smaller flag.

2. TENTS Jana went camping and stayed in a tent shaped like a triangle. In a photo of the tent, the base of the tent is 6 inches and the altitude is 5 inches. The actual base was 12 feet long. What was the height of the actual tent?

x

3. PLAYGROUND The playground at Hank’s school has a large right triangle painted in the ground. Hank starts at the right angle corner and walks toward the opposite side along an angle bisector and stops when he gets to the hypotenuse.

A

B

45˚

30 ft

40 ft

50 ft

How much farther from Hank is point Bversus point A?

4. FLAG POLES A flag pole attached to the side of a building is supported with a network of strings as shown in the figure.

A BC

DF

E

The rigging is done so that AE = EF, AC = CD, and AB = BC. What is the ratio of CF to BE?

5. COPIES Gordon made a photocopy of a page from his geometry book to enlarge one of the figures. The actual figure that he copied is shown below.

39 mm

Med

ian

Altit

ude

30 mm

29 mm

The photocopy came out poorly. Gordon could not read the numbers on the photocopy, although the triangle itself was clear. Gordon measured the base of the enlarged triangle and found it to be 200 millimeters.

a. What is the length of the drawn altitude of the enlarged triangle? Round your answer to the nearest millimeter.

b. What is the length of the drawn median of the enlarged triangle? Round your answer to the nearest millimeter.

7-5

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Solve for x.

1)

x

2)

x

3)

x

4)

x

5)

x

6)

x

7)

x

8)x

-1-

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Regents Exam Questions G.G.46: Side Splitter Theorem Name: ________________________ www.jmap.org

1

G.G.46: Side Splitter Theorem: Investigate proportions among segments of sides of the triangle, given line(s) parallel to one side and intersecting the other sides of the triangle

1 In the accompanying diagram of equilateral triangle ABC, DE 5 and DE AB.

If AB is three times as long as DE, what is the perimeter of quadrilateral ABED?1) 202) 303) 354) 40

2 In ABC, point D is on AB, and point E is on BC such that DE AC. If DB 2, DA 7, and DE 3, what is the length of AC?1) 82) 93) 10.54) 13.5

3 In the diagram below of ACD, E is a point on AD and B is a point on AC, such that EB DC . If AE 3, ED 6, and DC 15, find the length of EB.

4 In the diagram below of ADE, B is a point on AE and C is a point on AD such that BC ED, AC x 3, BE 20, AB 16, and AD 2x 2. Find the length of AC.

Page 19 of 24

Geometry Practice: G.G.46www.jmap.org NAME:_____________________________

P.I. G.G.46: Investigate, justify, and apply theoremsabout proportional relationships among the segmentsof the sides of the triangle, given one or more linesparallel to one side of a triangle and intersecting theother two sides of the triangle

1. Given: PQ BC || . Find the length of AB .

4 6

15

P Q

B C

A

[A] 14 [B] 11 [C] 16 [D] 18

2. In the figure shown, BC || DE , AB = 2 yards,BC = 9 yards, AE = 36 yards, andDE = 36 yards. Find BD .

D E

A

CB

[A] 9 yd [B] 8 yd [C] 6 yd [D] 27 yd

3. In the figure shown, ABC ADE, AB = 7yards, BC = 8 yards, AE = 4 yards, and DE = 16yards. Find CE.

D E

A

CB

4. Given: PQ BC || . Find the measure of CQ .

6

12

9P Q

B C

A

5. Given AE || BD , solve for x.

AB

CD

E82

x

10

6. Given AE || BD , solve for x.

A

B

CD

E63

x

8

7. Find x.

10 14

x 21

Page 20 of 24

Regents Exam Questions G.G.42: Midsegments Name: ________________________ www.jmap.org

1

G.G.42: Midsegments: Investigate, justify, and apply theorems about geometric relationships, based on properties of the segment joining the midpoints of two sides of the triangle

1 If the midpoints of the sides of a triangle are connected, the area of the triangle formed is what part of the area of the original triangle?

1) 14

2) 13

3) 38

4) 12

2 In the diagram below of ABC, D is the midpoint of AB, and E is the midpoint of BC .

If AC 4x 10, which expression represents DE?1) x 2.52) 2x 53) 2x 104) 8x 20

3 In the diagram below of ABC, DE is a midsegment of ABC, DE 7, AB 10, and BC 13. Find the perimeter of ABC.

4 In the diagram of ABC below, AB 10, BC 14, and AC 16. Find the perimeter of the triangle formed by connecting the midpoints of the sides of

ABC.

Page 21 of 24

Regents Exam Questions G.G.42: Midsegments Name: ________________________ www.jmap.org

2

5 In the diagram below of ACT , D is the midpoint of AC, O is the midpoint of AT , and G is the midpoint of CT .

If AC 10, AT 18, and CT 22, what is the perimeter of parallelogram CDOG?1) 212) 253) 324) 40

6 On the set of axes below, graph and label DEF with vertices at D( 4, 4), E( 2,2), and F(8, 2). If G is the midpoint of EF and H is the midpoint of DF , state the coordinates of G and H and label each point on your graph. Explain why GH DE .

Page 22 of 24


P.I. G.G.42: Investigate, justify, and applytheorems about geometric relationships, based onthe properties of the line segment joining themidpoints of two sides of the triangle

1. Solve for x given BD = 52

3x and AE =

6 4x . Assume B is the midpoint of ACand D is the midpoint of CE.

A

B

C

D

E

[A] 2 [B] 29

[C] 12

[D] 92

2. Solve for x given BD = 72

2x and AE =

3 6x . Assume B is the midpoint of ACand D is the midpoint of CE.

A

B

C

D

E

[A] 72

[B] 12

[C] 27

[D] 2

3. Solve for x given BD = 5 2x and AE =9 6x . Assume B is the midpoint of ACand D is the midpoint of CE.

A

B

C

D

E

4. Solve for x given BD = 4 2x and AE =6 8x . Assume B is the midpoint of ACand D is the midpoint of CE.

A

B

C

D

E

Page 23 of 24


5. Find the area of the rectangle if AC = 11and BD = 22.

A

B

C

D

[A] 121 [B] 33 [C] 242 [D] 60.5


A

B

C

D

[A] 39 [B] 360 [C] 180 [D] 90


A

B

C

D


A

B

C

D

9. The vertices of a triangle are A( 3 , 2),B(3, 4), and C(1, 6 ). Find thecoordinates of S, the midpoint of AB, andT, the midpoint of BC. Verify that

ST AC12

and that ST AC|| .

10. Find the values of x and y.

y6 4x

x

4 7x

[A] x y3 12

25,

[B] x y3 12

12 12

,

[C] x y3 12

24,

[D] none of the above

Page 24 of 24

Kuta Solving ProportionsFormal Skills Practice 7-1 Ratios and ProportionsFormal Practice 7-1 Ratios and ProportionsFormal Skills Practice 9-6 DilationsFormal Practice 9-6 DilationsFormal Word Problems 9-6 DilationsOther Dilations 7JMAP Dilations3Formal Skills Practice 7-2 Similar PolygonsFormal Practice 7-2 Similar PolygonsFormal Skills Practice 7-3 Similar TrianglesFormal Practice 7-3 Similar TrianglesFormal Word Problems 7-3 Similar TrianglesJMAP Similarity of triangles1JMAP Similarity of triangles3JMAP Similarity of triangles5Formal Skills Practice 7-5 Parts of Similar TrianglesFormal Practice 7-5 Parts of Similar TrianglesFormal Word Problems 7-5 Parts of Similar TrianglesFormal Skills Practice 7-4 Parallel Lines and Proportional PartsFormal Practice 7-4 Parallel Lines and Proportinal PartsKuta Solving Triangles with Proportional PArtsJMAP Triangles divided by parrallel lines1JMAP Triangles divided by parrallel lines2JMAP Midsegment Theorem1JMAP Midsegment Theorem2A Proof of the Pythagorean TheoremBlank Page

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