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”

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’(4  2, 2  2)  A’(8, 4) B(2, 0)  B’(2  2, 0  2)  B’(4, 0) C(6, -6)  C’(6  2, -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)

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.

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.

C C is the center of the dilation mapping Δ XYZ onto Δ LMN Y X Z N M L The center of any dilation is where the lines through all corresponding points intersect.

Dilation types Contractionreduction Contraction: reduction: the image is smaller than the preimage: scale factor is greater than 0, but less than 1. Expansionenlargement 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 3 Triangle 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 YZ AB BC CD = = = W A X Y BC

Example 1 D F E B 80° A C 40° If Δ ABC is similar to Δ DEF 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!

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 y x y - 3 A BC D E F

Example 4 In the figure below, Δ ABC is similar to Δ DEF. What is the length of DE? A.12 B.11 C.10 D.7 ⅓ E.6 ⅔ A 10 B 11 C 12 DF8 E