Dilations Determine whether a dilation is an enlargement

Dilations • 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) B’ r=2 A B center C D D’

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. Another example: XM’ = ⅓(XM) r = 1/3 M N XN’ = ⅓(XN) XO’ = ⅓(XO) XP’ = ⅓(XP) P M’ P’ O N’ O’ X center

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 C and B D D’ M P N M’ P’ O N’ O’

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 E C

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 b. A’B’ = 36, r = ¼ A’B’ = |r|(AB) A’B’ = 2(12) 36 = ¼(AB) A’B’ = 24 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 C J L L’ J’ K’

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 -4 -6 -8 -10 -12 5 10 15

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. Preimage (x, y) Image (2 x, 2 y) A(7, 10) A’(14, 20) B(-4, 6) B’(8, -12) C(-2, 3) C’(-4, 6) 22 20 18 16 14 12 10 8 6 4 2 -10 -5 -2 -4 -6 -8 -10 -12 5 10 15

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 A’ B’ 8 Scale factor = image length preimage length B A 7 6 C 5 = 6 units 3 units 4 D 3 2 =2 E D’ E’ 1 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 Scale factor = image length preimage length H G 7 6 C 5 = 4 units 4 J 3 F 2 =1 1 2 4 The dilation is a congruence transformation 6 8 10 12