Connection to previews lesson…

Previously, we studied rigid transformations, in which the image

and preimage of a figure are congruent. In this lesson, you will

study a type of nonrigid transformation called a dilation, in which the image and preimage of a

figure are similar.

Standard: MCC9-12.G.SRT.1 Verify experimentally the properties of dilations

given by a center and a scale factor.EQ: What is a dilation and how does this

transformation affect a figure in the coordinate plane?

Dilations

Graph:

A(4, 2)

B(2, 0)

C(6, -6)

D(0, -4)

E(-6, -6)

F(-2, 0)

G(-4, 2)

H(0, 4)

Connect and label “original”.

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-5

5

10

10 155

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-5

5

10

10 155

A

B

C

D

E

F

G

H

When dilating a figure you need to have a scale factor.

For our first dilation use a scale factor of 2.

This means you will multiply each coordinate by 2 to get the new location.

A(4, 2) A’(42, 2 2) A’(8, 4)

B(2, 0) B’(22, 0 2) B’(4, 0)

C(6, -6) C’(62, -6 2) C’(__, __)

D(0, -4)

E(-6, -6)

F(-2, 0)

G(-4, 2)

H(0, 4)

Graph the dilation with a scale factor of 2:

A’(8, 4)

B’(4, 0)

C’(12, -12)

D’(0, -8)

E’(-12, -12)

F’(-4, 0)

G’(-8, 4)

H’(0, 8)

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5

10

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A

B

C

D

E

F

G

H

H’

A’

B’

C’

D’

E’

F’

G’

A(4, 2) A”( , )

B(2, 0) B”( , )

C(6, -6)

D(0, -4)

E(-6, -6)

F(-2, 0)

G(-4, 2)

H(0, 4)

Here are the original points…

Now on your graph paper calculate the coordinates for a dilation with a scale

factor of 0.5.

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A

B

C

D

E

F

G

H

H’

A’

B’

C’

D’

E’

F’

G’

H’’A’’

B’’

C’’

D’’

E’’

F’’G’’

Vocabulary:

Dilation: Transformation that changes the size of a figure, but not

the shape.

Scale factor: The ratio of any 2 corresponding lengths of the sides of 2

similar figures.

Corresponding Sides: Sides that have the same relative positions in geometric

figures.

Vocabulary:Congruent: Having the same size, shape and measure. 2 figures are

congruent if all of their corresponding measures are equal.

Congruent figures: Figures that have the same size and shapes.

Corresponding Angles: Angles that have the same relative positions in geometric

figures.

Vocabulary:

Parallel Lines: 2 lines are parallel if they lie in the same plane and

do not intersect.

Proportion: An equation that states that 2 ratios are equal.

Ratio: Comparison of 2 quantities by division and may be

written as r/s, r:s, or r to s.

Vocabulary:

Transformation: The mapping or movement of all points of a

figure in a plane according to a common operation.

Similar Figures: Figures that have the same shape but not

necessarily the same size.

Dilation properties•When dilating a figure in a coordinate plane, a segment in the original image (not passing through the center), is parallel to it’s corresponding segment in the dilated image.

•When given a scale factor, the corresponding sides of the dilated image become larger of smaller by the scale factor ratio given.

CC is the center of the dilation mapping ΔXYZ onto ΔLMNY

X Z

N

M

L

The center of any dilation is where the lines through all corresponding points intersect.

Dilation types

Contraction: reduction: the image is smaller than the preimage: scale factor is greater than 0, but less than 1.

Expansion: enlargement: the image is larger than preimage: Scale factor is greater than 1.

Example 1

A picture is enlarged by a scale factor of 125% and then enlarged again by the same scale factor. If the original picture was 4” x 6”, how large is the final copy?

By what scale factor was the original picture enlarged?

Example 2

A triangle has coordinates A(3,-1), B(4,3) and C(2,5). The triangle will undergo a dilation using a scale factor of 3. Determine the coordinates of the vertices of the resulting triangle.

Example 3Triangle ABC is a dilation of triangle XYZ. Use the coordinates of the 2 triangles to determine the scale factor of the dilation.

A(-1, 1), B(-1, 0), C(3,1)X(-3, 3), Y(-3, 0), Z(9, 3)

Similar Figures

Two figures, F and G, are similar (written F ~ G) if and only if

a.) corresponding angles are congruent and

b.)corresponding sides are proportional.

Dilations always result in similar figures!!!

Similar Figures

If WXY ~ ABC, then:∠W ≅ ∠A ∠X ≅ ∠B

∠Y ≅ ∠C

WX XY YZAB BC CD

= = =

W

A

X Y

B C

Example 1

D F

E B80°

A C40°

If ΔABC is similar to ΔDEF in the diagram below, then m∠D = ?

A.80°B.60°C.40°D.30°E.10°

Example 2

Determine whether the triangles are similar. Justify your response!

129

5

3.75

13 9.75

Example 3

Triangle ABC is similar to triangle DEF. Determine the scale factor of DEF to ABC (be careful – the order is important), then calculate the lengths of the unknown sides.

12

15

y + 3

9 x

y - 3

A

B C

D

E F

Example 4In the figure below, ΔABC is

similar to ΔDEF. What is the length of DE?A. 12B. 11C. 10D. 7⅓E. 6⅔

A

10

B

11

C12 D F8

E