UNIT 1 SIMILARITY, CONGRUENCE, AND PROOFS Lesson 1: Investigating Properties of Dilations Instruction
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1 Prerequisite Skills This lesson requires the use of the following skills: operating with fractions, decimals, and percents converting among fractions, decimals, and percents Introduction A figure is dilated if the preimage can be mapped to the image using a scale factor through a center point, usually the origin. You have been determining if figures have been dilated, but how do you create a dilation? If the dilation is centered about the origin, use the scale factor and multiply each coordinate in the figure by that scale factor. If a distance is given, multiply the distance by the scale factor. Key oncepts The notation is as follows: Dk ( x, y) = ( kxky, ). Multiply each coordinate of the figure by the scale factor when the center is at (0, 0). y k D = D D (x, y) D (kx, ky) x The lengths of each side in a figure also are multiplied by the scale factor. U1-31
2 If you know the lengths of the preimage figure and the scale factor, you can calculate the lengths of the image by multiplying the preimage lengths by the scale factor. Remember that the dilation is an enlargement if k > 1, a reduction if 0 < k < 1, and a congruency transformation if k = 1. ommon Errors/Misconceptions not applying the scale factor to both the x- and y-coordinates in the point improperly converting the decimal from a percentage missing the connection between the scale factor and the ratio of the image lengths to the preimage lengths U1-32
3 Guided Practice Example 1 If AB has a length of 3 units and is dilated by a scale factor of 2.25, what is the length of AB? Does this represent an enlargement or a reduction? 1. To determine the length of AB, multiply the scale factor by the length of the segment. AB = 3; k = 2.25 A'B' = k AB A'B' = = 6.75 AB is 6.75 units long. 2. Determine the type of dilation. Since the scale factor is greater than 1, the dilation is an enlargement. Example 2 A triangle has vertices G (2, 3), H ( 6, 2), and J (0, 4). If the triangle is dilated by a scale factor of 0.5 through center (0, 0), what are the image vertices? Draw the preimage and image on the coordinate plane. 1. Start with one vertex and multiply each coordinate by the scale factor, k. D k = (kx, ky) G' = D 0.5 [G (2, 3)] = D 0.5 (0.5 2, 0.5 3) = (1, 1.5) 2. Repeat the process with another vertex. Multiply each coordinate of the vertex by the scale factor. H' = D 0.5 [H ( 6, 2)] = D 0.5 (0.5 6, 0.5 2) = ( 3, 1) U1-33
4 3. Repeat the process for the last vertex. Multiply each coordinate of the vertex by the scale factor. J' = D 0.5 [ J (0, 4)] = D 0.5 (0.5 0, 0.5 4) = (0, 2) 4. List the image vertices. G' (1, 1.5) H' ( 3, 1) J' (0, 2) 5. Draw the preimage and image on the coordinate plane. y H ( 6, 2) H ( 3, 1) G (1, 1.5) -2-3 G (2, 3) J (0, 4) J (0, 2) x U1-34
5 Example 3 What are the side lengths of D E with a scale factor of 2.5 given the preimage and image below and the information that DE = 1, E = 9.2, and D = 8.6? D E D 1 E hoose a side to start with and multiply the scale factor (k) by that side length. DE = 1; k = 2.5 D'E' = k DE D'E' = = hoose a second side and multiply the scale factor by that side length. E = 9.2; k = 2.5 E'' = k E E'' = = 23 U1-35
6 3. hoose the last side and multiply the scale factor by that side length. D = 8.6; k = 2.5 'D' = k D 'D' = = Label the figure with the side lengths. D 2.5 E D 1 E U1-36
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