Dilations or a congruence transformation. Determine whether a dilation is an enlargement, a reduction, or a congruence transformation. Determine the scale factor for a given dilation. Vincent Van Gogh’s “Starry Night” reduced by a factor of 2

CLASSIFY DILATIONS A dilation is a transformation that may change the size of a figure. A dilation requires a center point and a scale factor.

CLASSIFY DILATIONS A dilation is a transformation that may change the size of a figure. A dilation requires a center point and a scale factor. Example: A’ CA’ = 2(CA) CB’ = 2(CB) CD’ = 2(CD) r = 2 B’ A B center C D D’

CLASSIFY DILATIONS r = 1/3 A dilation is a transformation that may change the size of a figure. A dilation requires a center point and a scale factor. Another example: r = 1/3 XM’ = ⅓(XM) XN’ = ⅓(XN) XO’ = ⅓(XO) XP’ = ⅓(XP) M N O P M’ N’ center O’ P’ X

Key Concept Dilation If |r| > 1, the dilation is an enlargement. If 0 < |r| < 1, the dilation is a reduction. If |r| = 1, the dilation is a congruence transformation.

Key Concept Dilation We have seen that dilations produce similar figures, therefore dilation is a similarity transformation. A’ B’ A B C D D’ and M N P O M’ N’ P’ O’ X

Theorem If a dilation with center C and a scale factor of r transforms A to E and B to D, then ED = |r|(AB) A D B C E

Example 1 Determine Measures Under Dilations Find the measure of the dilation image A’B’ or the preimage AB using the given scale factor. a. AB = 12, r = -2 A’B’ = |r|(AB) A’B’ = 2(12) A’B’ = 24 b. A’B’ = 36, r = ¼ A’B’ = |r|(AB) 36 = ¼(AB) AB = 144 When the scale factor is negative, the image falls on the opposite side of the center than the preimage.

Key Concept Dilations If r > 0, P’ lies on CP, and CP’ = r ∙ CP If r < 0, P’ lies on CP’ the ray opposite CP, and CP’ = r ∙ CP The center of dilation is always its own image.

Example 2 Draw a Dilation Draw the dilation image of ∆JKL with center C and r = ½ K J L

Example 2 Draw a Dilation Draw the dilation image of ∆JKL with center C and r = -½ K Since 0 < |r| < 1, the dilation is a reduction J L K L’ C J’ J K’ L

Theorem If P(x, y) is the preimage of a dilation centered at the origin with scale factor r, then the image is P’(rx, ry).

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image. 22 20 18 16 14 12 10 8 6 4 2 -10 -5 -2 5 10 15 -4 -6 -8 -10 -12

Example 3 Dilations in the Coordinate Plane COORDINATE GEOMETRY Triangle ABC has vertices A(7, 10), B(4, -6), and C(-2, 3). Find the image of ∆ABC after a dilation centered at the origin with the scale factor of 2. Sketch the preimage and the image. 22 20 18 16 Preimage (x, y) Image (2x, 2y) A(7, 10) A’(14, 20) B(-4, 6) B’(8, -12) C(-2, 3) C’(-4, 6) 14 12 10 8 6 4 2 -10 -5 -2 5 10 15 -4 -6 -8 -10 -12

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a. 9 8 7 6 5 4 3 2 1 A’ B’ Scale factor = image length preimage length = 6 units 3 units = 2 A B C E D E’ D’ 2 4 6 8 10 12

Example 4 Identify the Scale Factor Determine the scale factor for each dilation with center C. Then determine whether the dilation is an enlargement, reduction, or congruence transformation. a. 9 8 7 6 5 4 3 2 1 Scale factor = image length preimage length = 4 units 4 units = 1 G H C J F 2 4 6 8 10 12 The dilation is a congruence transformation