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Exploring Dilations and Similarity in Geometry

This guide covers the concepts of dilations with scale factors both greater than and less than one, including the verification of similarity between figures. Use examples involving quadrilaterals and triangles to demonstrate how to draw dilations by multiplying vertex coordinates. The process includes sketching original figures and their dilated images, confirming proportionality of corresponding sides, and applying the SAS Similarity Theorem for validation. Practice problems are included to reinforce the learning experience.

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Exploring Dilations and Similarity in Geometry

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  1. (x, y) (2x, 2y) A(2, 1) L(4, 2) B(4, 1) M(8, 2) C(4, –1) N(8, –2) D(1, –1) P(2, –2) EXAMPLE 1 Draw a dilation with a scale factor greater than 1 Draw a dilation of quadrilateralABCDwith verticesA(2, 1), B(4, 1), C(4, – 1), andD(1, – 1). Use a scale factor of2. SOLUTION First draw ABCD. Find the dilation of each vertex by multiplying its coordinates by 2. Then draw the dilation.

  2. A triangle has the vertices A(4,– 4),B(8, 2), and C(8,– 4). The image of ABCafter a dilation with a scale factor of is DEF. SketchABCandDEF. Verify that ABCand DEFare similar. 1 a. 2 b. EXAMPLE 2 Verify that a figure is similar to its dilation

  3. a. The scale factor is less than one, so the dilation is a reduction. 1 1 2 2 A(4, – 4) D(2, – 2) y x, (x, y) C(8, – 4) F(4, – 2) B(8, 2) E(4, 1) EXAMPLE 2 Verify that a figure is similar to its dilation SOLUTION

  4. b. Because Cand Fare both right angles, C F. Show that the lengths of the sides that include Cand Fare proportional. Find the horizontal and vertical lengths from the coordinate plane. ? = AC BC = 4 6 EF DF So, the lengths of the sides that include Cand Fare proportional. 2 3 ANSWER Therefore, ABCDEFby the SAS Similarity Theorem. ~ EXAMPLE 2 Verify that a figure is similar to its dilation

  5. Find the coordinates of L, M, and N so that LMN is a dilation of PQR with a scale factor of k. Sketch PQR and LMN. x, y = 4x , 4y P (–2, –1) = L (–8, –4) M (– 4, 0) Q (–1, 0) = R (0, –1) = N (0, –4) for Examples 1 and 2 GUIDED PRACTICE 1. P(–2, 21), Q(–1, 0), R(0, –1); k = 4 SOLUTION Find the dilation of each vertex by multiplying its coordinates by 4. ANSWER

  6. 2 2 5 5 x, y x , y P(5, –5) L (2, –2) Q(10,–2) M (4, –2) R (10, 5) N (4, 2) for Examples 1 and 2 GUIDED PRACTICE 2. P(5, –5), Q(10, –5), R(10, 5); k = 0.4 SOLUTION Find the dilation of each vertex by multiplying its coordinates by 0.4. The scale is less than one, so the dilation is a reduction. ANSWER

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