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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. Dilations 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? Graph: 10 A(4, 2) B(2, 0) 5 C(6, -6) D(0, -4) -15 -10 -5 5 E(-6, -6) F(-2, 0) -5 G(-4, 2) H(0, 4) Connect and label “original”. -10 10 15 10 5 H A G B F -15 -10 -5 5 -5 E 10 D C -10 15 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) 10 H’ 5 G’ H A’ A G F’ -10 -15 B F -5 B’ 5 10 15 D -5 E C D’ -10 E’ C’ Now on your graph paper calculate the coordinates for a dilation with a scale factor of 0.5. Here are the original points… 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) 10 H’ 5 G’ G’’ F’ -10 A’ A G -15 H F H’’ A’’ F’’ B’’ B -5 B’ 5 10 15 D’’ E’’ C’’ D -5 E C D’ -10 E’ C’ 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. The center of any dilation is where the lines through all corresponding points intersect. L C C is the center of the dilation mapping ΔXYZ onto ΔLMN Y X Z M N 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 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!!! W Similar Figures If WXY ~ ABC, then: ∠W ≅ ∠A ∠X ≅ ∠B Y X ∠Y ≅ ∠C A WX XY YZ AB BC CD = = = B C Example 1 If ΔABC is similar to ΔDEF in the diagram below, then m∠D = ? A.80° B. 60° C.40° D.30° E. 10° B E 80° D F 40° A C Example 2 Determine whether the triangles are similar. Justify your response! 13 12 9 9.75 3.75 5 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. A 12 B D 15 y+3 9 C E y-3 x F Example 4 In the figure below, ΔABC is similar to ΔDEF. What is the length of DE? A. 12 E B B. 11 11 10 C. 10 C D. 7⅓ A D 8 12 E. 6⅔ F