Chapter 7 Similarity and Proportion Express a ratio in simplest form. State and apply the properties...

Chapter 7Similarity and Proportion

• Express a ratio in simplest form.

• State and apply the properties of similar polygons.

• Use the theorems about similar triangles.

7.1 Ratio and Proportion

Objectives

• Express a ratio in simplest form

• Solve for an unknown in a proportion

Ratio

• A comparison between numbers

12

35

69

ab

5 : 7

5 : 7 : s : 5 : t

Ratio

• Always reduce ratios to the simplest form

Proportion

• An equation containing ratios

5

8

a

b

3

5

a

b

3

5 15

a

3 4

5 2

a d

b c

Solving a Proportion

3

5 15

a First, cross-multiply

5 45a Next, divide by 5

9a

White Board Practice

• ABCD is a parallelogram. Find the value of each ratio.

A

D C

B

10

6


• AB : BC

A

D C

B

10

6


• 5 : 3

A

D C

B

10

6


• BC : AD

A

D C

B

10

6


• 1 : 1

A

D C

B

10

6


• m A : m C

A

D C

B

10

6


• 1 : 1

A

D C

B

10

6


• AB : perimeter of ABCD

A

D C

B

10

6


• 5 : 16

A

D C

B

10

6


• x = 2 and y = 3. Write each ratio in simplest form.

x to y



2 to 3



6x2 to 12xy



1 to 3



y – x

x



1

2

7.2 Properties of Proportions

Objectives

• Express a given proportion in an equivalent form.

Means and Extremes

• The extremes of a proportion are the first and last terms• The means of a proportion are the middle terms

=a

b

c

da : b = c : d

Properties of Proportion

a c

b d is equivalent to

a b

c dad bc b d

a c

a b c d

b d

1.

4.

3. 2.

That just means that you can rewrite

5

2

x

yyx 52

2

25

y

yx

As any of these

1.

4.

3. 2. 25

yx

2

5

y

x

Another Property

...If ... then

...

a c e a a c e

b d f b b d f


• If , then 2x = _______2

4

7

x


• If , then 2x = 282

4

7

x


• If 2x = 3y, then 3

2


• If 2x = 3y, then x

y

3

2


• If , then2

4

7

x

7

7x


• If , then2

4

7

x

2

6

7

7

x


• If , then2

2

3

yx 3

3x


• If , then2

2

3

yx23

3 yx

7.3 Similar Polygons

Objectives

• State and apply the properties of similar polygons.

Similar Polygons

• Same shape

• Not the same size Why?

• Not the same size Why?

Because then they would be congruent !

Similar Polygons (~)• All corresponding angles congruent

A A’

B B’

C C’A

B

C

A’

B’ C’

Similar Polygons (~)

• All corresponding sides in proportion

AB = BC = CA

A’B’ B’C’ C’A’A

B C

A’

B’ C’

The Scale Factor

The reduced ratio between any pair of corresponding sides or the perimeters.

12:3

12

3

Finding Missing Pieces

You have to know the scale factor first to find missing pieces.

12

3

y10

y

10

3

12


• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find their scale factor

A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30


• 5:3

A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30


• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the values of x, y, and z

A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30


• x = 18

• y = 20

• z = 13.2A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30


• Quadrilateral ABCD ~ Quadrilateral A’B’C’D’. Find the ratio of the perimeters

A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30


• 5:3

A

D C

B

A’

D’ C’

B’

50

y

30 20 12

x z

30

7.4 A Postulate for Similar Triangles

Objectives

• Learn to prove triangles are similar.

AA Simliarity Postulate(AA~ Post)

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.

A

B C

D

EF

Remote Time

• T – Similar Triangles

• F – Not Similar

T – Similar TrianglesF – Not Similar

7-5: Theorems for Similar Triangles

Objectives

• More ways to prove triangles are similar.

SAS Similarity Theorem (SAS~)

If an angle of a triangle is congruent to an angle of another triangle and the sides including those angles are proportional, then the triangles are similar. A

B C

D

EF

SSS Similarity Theorem (SSS~)

If the three sides of one triangle are proportional to the three sides of another triangle, then the triangles are similar.

A

B C

D

EF

Homework Set 7.5

• 7-5 #1-19 odd

• WS PM 40

7-6: Proportional Lengths

Objectives

• Apply the Triangle Proportionality Theorem and its corollary

• State and apply the Triangle Angle-bisector Theorem

Divided ProportionallyIf points are placed on segments AB and CD so that , then we say that these

segments are divided proportionally.AX CY

XB YD

A

X

B

C

Y

D

See It!

Theorem 7-3If a line parallel to one side of a triangle intersects the

other two sides, it divides them proportionally.

X

Y

Z

A B

See It!

CorollaryIf three parallel lines intersect two transversals,

they they divide the transversal proportionally.

R

S

T

W

X

Y

See It!

Theorem 7-4If a ray bisects an angle of a triangle, then it divides the

opposite side into segments proportional to the other two sides.

X

Y

Z

W

See It!

Construction 12Given a segment, divide the segment into a given number of congruent segments.

Given:

Construct:

Steps:

AB 1a segment of length

3CD AB

Construction 13Given three segments, construct a fourth segment so that the four segments are proportional.

Given:

Construct:

Steps:

, , AB CD EF AB EF so that

CD XYXY

Homework Set 7.6

• WS PM 41

• WS Constructions 12 and 13

• 7-6 #1-23 odd

• Quiz next class day

Chapter 7 Similarity and Proportion Express a ratio in simplest form. State and apply the properties...

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