Lesson 8.1 & 8.2Solving Problems with Ratio and Proportion

Today, we will learn to……find and simplify ratios...use proportions to solve problems

RatioA ratio is a comparison of

two numbers written in simplest form.

a a : b a to b b

Simplify the ratio.K H D M D C M

1. 2 m : 300 cm200 cm : 300 cm 2 : 3

2. 2 km : 600 m 2000 m : 600 m 10 : 3

3. 10 mm : 5.5 cm 10 mm : 55 mm 2 : 11

4. In the diagram, DE : EF is1 : 2 and DF = 45. Find DE and EF.

D E F

1x + 2x = 453x = 45x = 15

1 2

DE =EF =

x x

1530

5. In ΔABC, the measures of the angles are in the

extended ratio of 3:4:5. Find the measures of the angles.

12x = 180x = 15

3x + 4x + 5x = 180

°, °, °

What do we know about the angles of a triangle?

45 60 75

6. The perimeter of a rectangle is 70 cm. The ratio of the length to the width is 3 : 2. Find the length and the width of the rectangle.

3x

2x 2x

3x 3x+2x+3x+2x = 70

Length is Width is

10x = 70x = 7

2114

7. A triangle has an area of 48 m 2. The ratio of the base to the

height is 2 : 3. Find the base and height.

A = ½ bh48 = ½ (2x)(3x)48 = 3x2

16 = x2

4 = xbase is height is

8 m12 m

2x

3x

Solve the proportion for x.

8. 2 8 7 x-2

2(x-2) = 562x - 4 = 56

2x = 60x = 30

9. On a map, 2 inch = 180 miles. Two cities are about 2 ¾ inches apart.

Estimate the actual distance between them.

2 in 180 mi

2x = 180(2¾) x = 247.5 miles

2 ¾ inx mi

10. In a photograph taken from an airplane, a section of a city street is 3 1/2 inches long and

1/8 of an inch wide. If the actual street is 30 feet wide, how long is it?

1/8 30x=

x = 840 feet

31/2

x = (3 )(30)1/8 1/2

11. AB : AC is 3 : 2. Find x.

3x+3x+1 2=

3(x+1) = 2(x+3)3x+3 = 2x+6

x + 3 = 6x = 3

12. Given MN MP find PQ. NO PQ=

x

14-x

46

14-xx=

x = 8.4

4x = 6(14-x)

?

?

4x = 84 - 6x4

M

6

N

O Q

P14

5

A

2B

C E

D

7+x

x

7

57

77+x=

x = 2.8

Given AB AD find DE. AC AE=13.

5(7+x) = 49

?

? 35+5x = 495x = 14

14. Standard paper sizes are all over the world. The sizes all have the same width-to-length ratios. Two sizes of paper shown are A4 and A3. Find x.

210 mm

x

x

420 mm

210 x420x =

x2 = (210)(420)

x ≈ 297 mm

15. The batting average of a baseball player is the ratio of

the number of hits to the number of official at-bats.

x.308643 1= x = (643)(.308)

x = 198 hits

In 1998, Sammy Sosa of the Chicago Cubs had 643 official at-bats and a batting average of .308. How many hits did Sammy Sosa get?

16. A wheelchair ramp should have a slope of 1/12. If a ramp rises 2

feet, what is its run?

1 2 ftx12 =

x = (12)(2 ft)x = 24 feet

What is its length? length2 = 22 + 242 length2 = 4 + 576 length2 = 580 length = 24.08 feet

2 ft ?

Geometric MeanThe geometric mean of two

positive numbers

(a and b) is …. a x

x b

Find the geometric mean of the given numbers.

35 and 175

x ≈ 78.3

xx35

175=

x2 = 35(175)

Lesson 8.3Similar Polygons

Today, we will learn to……identify similar polygons...use similar polygons

Two polygons are similar ifall corresponding angles are congruent and corresponding

sides are proportional.

AB BC AC

ΔABC ~ Δ XYZ if

A B C X Y Z

XY YZ XZand

B

C

DA

FG

HE

ABCD ~ EFGH

CDGH

ADEH

ABEF

BCFG

A E, B F, C G, D H

Statement of Proportionality

Scale Factor

The scale factor is the ratio of the lengths of two corresponding sides.

6 8 10

1. Are the triangles similar? If they are, find the scale factor and write a

statement of similarity.

9 12 15

Yes, the scale factor is 2

3XAR ~ __ __ __

M N T

4.5 6 9

2. Are the triangles similar? If they are, find the scale factor and write a

statement of similarity.

6 8 12

Yes, the scale factor is 3

4LMN ~ __ __ __T P O

12 15 x

A

B

C

D

E

F10

1215

12 y

x

4. Δ ABC ~ Δ DEF

15 10

3 2=

y 10 12 x 12 = 15

10

x = 1812 y = 15

10 y = 8

Scale Factor?

The triangles are similar. Find x and y.5.

AC

8 x

12

Map the triangles to find corresponding sides.

B

E

DF

9

y

18B

A C

x

12

8

B

A C

x

12

8

9 y 18 x 12 8

9 x =

x = 4

18 8

y 12 = 18

8

y = 27

5 = 3

6. RSTU ~ LMNO. Find the following.

125mT =mS =55

x 2.4

x

x = 4

7. You have a 3.5 inch by 5 inch photo that you want to enlarge. You want the enlargement to be 16 inches wide. How long will it be?

3.5 16x5 =

3.5x = (16)(5)x = 22.9 ≈ 23 inches

A triangular work of art and the frame around it are similar equilateral triangles.

12 in.

16 in.9. Find the ratio of the perimeters. (artwork : frame)

34

34

3648

8. Find the ratio of the artwork to the

frame.

The rectangles are similar.

11. Find the ratio of the perimeters.

45

45

2227.5

10. Find the ratio of corresponding sides.

47

5

8.75

220275

Theorem 8.1If 2 polygons are similar,

then the ratio of the perimeters is __________ the ratio of corresponding

side lengths.

equal to

12. The patio around a pool is similar to the pool. The perimeter of the pool is 96 feet. The ratio of the

patio to the pool is 3 to 2. Find the perimeter

of the patio.

3 x962 = 2x = (3)(96)

x = 144 feet

patiopool

Turn to page 145 in your workbook!

Lesson 8.4

Proving Triangles are Similar Triangles

Today, we will learn to……identify similar triangles...use similar triangles

Postulate 25Angle-Angle (AA) Similarity

Two triangles are similar if 2 pairs of corresponding

angles are congruent.

Determine whether the triangles are similar. If they are, write a similarity statement.

1. R M

NL

27˚

LT S

35˚

80˚

65˚

80˚

ΔRTS ~ Δ____M

35˚65˚

LN

Determine whether the triangles are similar. If they are, write a similarity statement.

2. G

H

JK

L 27˚

27˚

ΔGLH ~ Δ____G KJ

4. If the triangles are similar, write a similarity statement.

31˚

47˚

not similar

5. If the triangles are similar, write a similarity statement.

43˚

not similar

6. The triangles are similar, find x.

3 5 7

x2 =3x = 10

x ≈ 3.33

yx2

53y2

3y = 14y ≈ 4.67

73

8. The triangles are similar. Find x. A B

C

D E

15

2518

9

x

15 925 x=

x = 15

Are the triangles similar? If they are, write a similarity statement.

Not ~ XZW ~ XTY

T

YX

X

Z

W

Are the triangles similar? If they are, write a similarity statement.

Not ~ABD ~ BCE

40

75

Lesson 8.5Proving Triangles are

Similar Triangles

Today, we will learn to……use similarity theorems to prove

that two triangles are similar

Theorem 8.2Side-Side-Side (SSS)

Similarity

If all three corresponding sides are proportional, then

the triangles are similar.

Determine whether the triangles are similar. If they are, write a

similarity statement.1. D

E

F8 10

12

A

B

C 1512

18

ΔACB ~ Δ____ by _____ DFE

12 15 188 10 12

scale factor?3:2

SSS

Theorem 8.3Side-Angle-Side (SAS)

Similarity If two sides are proportional

and the angles between them are congruent, then

the triangles are similar.

Determine if the triangles are similar. If they are, write a

similarity statement.2. A

B

C

D

E

F6

8

8

12

Not similar

8 12 6 8

Determine whether the triangles are similar. If they are, write a

similarity statement.3. A

B

C D

E5

5

3

3

ΔABE ~ Δ____ by _____ACD SAS

3 5 6 10Scale Factor? 1:2

Separate the triangles if it helps.3. A

C D

106

ΔACD ~ ΔABE by

B E

A

53

SAS

3 5 6 10

Find x. GLH ~ GKJ4.

x = 7.5

G

H

JK

L6 x

108 10 + x14

8 10 14 10 + x14 10 + x

8(10 + x) = 140

x = 7.5

G

H

JK

L6 x

108

8 10 6 x

8x = 60

What can we

conclude?

Find x. GLH ~ GKJ5.

Theorem 8.4Triangle Proportionality Theorem

If a line parallel to one side of a triangle intersects the other

two sides, then it divides the two sides proportionally.

Find x. The triangles are similar.

6. 7.

x12

14

39

x = 18 x = 4

12 x 5 10

x10

5

2

26 392 x

Estimate the height of the tree.8.

4 ft.

6 ft.

16 ft.

x ft.

x = 24 feet

4 6

=16 x

3 5.5

=

Estimate the height of the tree.9.

3 ft.

5.5 ft.

12 ft.

x ft.

x = 27.5 feet

15 x

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Lesson 8.6Proportions and Similar Triangles

Today, we will learn to……use proportionality theorems to

calculate segment lengths

Find the value of x.

1. 2.

x4

6

12

x = 4.8 x = 2.8 4 x

5 7

x7

5

2

10 12

2 x

Find the value of x.

3. 4. x5

7

15

x = 6.25 x ≈ 2.67

5 x

6 8

x8

6

2

2 x 12 15

Find the value of x.

5. 6.10

14 x

x = 21 x = 11

36 x

x 33

x20

10

24 14 10 30

3633

Theorem 8.5Triangle Proportionality

Converse

If a line divides two sides of a triangle proportionally, then it is

parallel to the third side.

Use similar triangles to find x.

7. 8.

12

7 8x

x = 6 x = 20 6 10

8 x

x10

86

?7 ?8

16 1212 x

Mid-Segment TheoremThe segment connecting the

midpoints of two sides of a triangle is parallel to the third

side and is _____ as long.half

Theorem 8.6

If three or more parallel lines intersect two transversals,

then they divide the transversals proportionally.

9. Find x and y.

y24

x 10.5

78

x= 10.5

24 =

y

x = 12

y = 21

8 7

78

10. Find x, y, and z.

x15

13 y

z10

30.4

15= x

30.438

38x = 456

x = 12

10. Find x, y, and z.

1215

13 y

z10

30.4

15= 12

y1315y = 156

y = 10.4

15= 12

z1015z = 120

z = 8x = 12

Theorem 8.7An angle bisector of a triangle divides the opposite side into segments whose lengths are proportional to the other two

sides.

11. Find x. 12. Find x.

21 =

24 x = 7

24

x 8

21

x 8

3

5

2x

3 =

2 x 5 x = 7.5

24

12

8

13. Find x.

?

8= 12

x 24-x

x = 9.624 - x

14. Find x and y.

18

16

8.5

What is another way to write y?

8.5 - x

18= 16

x 8.5-xx = 4.5

8.5 - x = 4

y = 4

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Lesson 8.7Dilations

Today, we will learn to……identify dilation...use properties of dilations to create a perspective

drawing

Dilation

A dilation is a transformation that

results in a reduction or enlargement of a figure.

=

1. A circle in a photocopier enlargement has a 6 inch diameter. If the enlargement percentage is 125%, what is the diameter of the preimage circle?

4.8 in.100 x125 6 in.

C3

6

Scale Factor = CPCP

'

P

Q

R

P´

Q´

R´Reduction

C

Scale Factor =

38

P

Q

R

P´

Q´

R´Reduction

38

new image preimage

C2

5P

QR

P´

Q´

R´

Scale Factor =

52

CPCP

'Enlargement

C

P

QR

P´

Q´R´

515

Scale Factor =

Enlargement new image preimage

C

P

Q

R

P´

Q´

R´

410

Find x.

x 6

x = 2.4 410 = x

6

2510 =

P

CQ

R

P´

Q´

R´

10

25Find x.

10

4

5

y

x

x = 25x

P

CQ

R

P´

Q´

R´

10

25Find y.

10

4

5

y

x

y = 102510 =

y4

Rectangle ABCD has vertices A (3,1) , B (3, 3) , C (2, 3), and D (2, 1). Find the coordinates of the dilation with center (0,0) and scale factor of 2.

Graph on next slide…

A (3,1) , B (3, 3) , C (2, 3), D (2, 1)

A’(6,2) , B’(6, 6) , C’(4, 6), D’(4, 2)

Scale Factor is 2

x

2x

A’

B’C’

D’

Do you notice a pattern?

Rectangle ABCD has vertices A (-3,3) , B (3, 6) , C (6, -3), and D (-3, -6). Find the coordinates of the dilation with center (0,0) and scale factor of 1/3.

(-1,1) B’D’ C’

A’ (2, -1)

(1, 2)(-1, -2)

A’ (-1,1) B’ (1,2) C’ (2,-1) D’(-1,-2)

Scale Factor is 1/3

A(-3,3) B(3, 6) C(6, -3) D(-3, -6)

A’B’

C’D’

A

B

C

D

Find x.

5126

=12x = 30

x = 2.5

x

A’B’C’

(0, 6)(6, 6)

(4.5, 3)

A’ B’

C’

ABC

(0, 4)(4, 4)(3, 2)

X’Y’Z’

(-0.75,-0.5)(2, 1)(1, -1)

XYZ

(-1.5, -1)(4, 2)(2, -2)X’

Y’

Z’

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