Dilations andand Similarity Dilations Similarity 7-6 7-6in the

Dilations andand Similarity Dilations Similarity 7-6 7-6in the Coordinate Plane in the Coordinate Plane Warm Up Lesson Presentation Lesson Quiz Holt Geometry Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Warm Up Simplify each radical. 1. 2. 3. Find the distance between each pair of points. Write your answer in simplest radical form. 4. C (1, 6) and D (–2, 0) 5. E(–7, –1) and F(–1, –5) Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Objectives Apply similarity properties in the coordinate plane. Use coordinate proof to prove figures similar. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Vocabulary dilation scale factor Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane A dilation is a transformation that changes the size of a figure but not its shape. The preimage and the image are always similar. A scale factor describes how much the figure is enlarged or reduced. For a dilation with scale factor k, you can find the image of a point by multiplying each coordinate by k: (a, b) (ka, kb). Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Helpful Hint If the scale factor of a dilation is greater than 1 (k 1), it is an enlargement. If the scale factor is less than 1 (k 1), it is a reduction. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 1: Computer Graphics Application Draw the border of the photo after a dilation with scale factor Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 1 Continued Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by Rectangle ABCD Holt Geometry Rectangle A’B’C’D’

Dilations and Similarity 7-6 in the Coordinate Plane Example 1 Continued Step 2 Plot points A’(0, 0), B’(0, 10), C’(7.5, 10), and D’(7.5, 0). Draw the rectangle. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 1 What if ? Draw the border of the original photo after a dilation with scale factor Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 1 Continued Step 1 Multiply the vertices of the photo A(0, 0), B(0, 4), C(3, 4), and D(3, 0) by Rectangle ABCD Holt Geometry Rectangle A’B’C’D’

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 1 Continued Step 2 Plot points A’(0, 0), B’(0, 2), C’(1.5, 2), and D’(1.5, 0). Draw the rectangle. 2 0 B’ A’ Holt Geometry C’ 1.5 D’

Dilations and Similarity 7-6 in the Coordinate Plane Example 2: Finding Coordinates of Similar Triangle Given that TUO RSO, find the coordinates of U and the scale factor. Since TUO RSO, Substitute 12 for RO, 9 for TO, and 16 for OY. 12OU 144 OU 12 Holt Geometry Cross Products Prop. Divide both sides by 12.

Dilations and Similarity 7-6 in the Coordinate Plane Example 2 Continued U lies on the y-axis, so its x-coordinate is 0. Since OU 12, its y-coordinate must be 12. The coordinates of U are (0, 12). So the scale factor is Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 2 Given that MON POQ and coordinates P (–15, 0), M(–10, 0), and Q(0, –30), find the coordinates of N and the scale factor. Since MON POQ, Substitute 10 for OM, 15 for OP, and 30 for OQ. 15 ON 300 ON 20 Holt Geometry Cross Products Prop. Divide both sides by 15.

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 2 Continued N lies on the y-axis, so its x-coordinate is 0. Since ON 20, its y-coordinate must be –20. The coordinates of N are (0, –20). So the scale factor is Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 3: Proving Triangles Are Similar Given: E(–2, –6), F(–3, –2), G(2, –2), H(–4, 2), and J(6, 2). Prove: EHJ EFG. Step 1 Plot the points and draw the triangles. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 3 Continued Step 2 Use the Distance Formula to find the side lengths. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 3 Continued Step 3 Find the similarity ratio. 2 Since 2 and E E, by the Reflexive Property, EHJ EFG by SAS . Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 3 Given: R(–2, 0), S(–3, 1), T(0, 1), U(–5, 3), and V(4, 3). Prove: RST RUV Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 3 Continued Step 1 Plot the points and draw the triangles. Y 5 4 U V 3 2 X S 1 -7 -6 -5 -4 -3 -2 -1 R Holt Geometry -1 T 1 2 3 4 5 6 7

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 3 Continued Step 2 Use the Distance Formula to find the side lengths. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 3 Continued Step 3 Find the similarity ratio. Since and R R, by the Reflexive Property, RST RUV by SAS . Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 4: Using the SSS Similarity Theorem Graph the image of ABC after a dilation with scale factor Verify that A'B'C' ABC. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 4 Continued Step 1 Multiply each coordinate by coordinates of the vertices of A’B’C’. Holt Geometry to find the

Dilations and Similarity 7-6 in the Coordinate Plane Example 4 Continued Step 2 Graph A’B’C’. B’ (2, 4) A’ (0, 2) C’ (4, 0) Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 4 Continued Step 3 Use the Distance Formula to find the side lengths. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Example 4 Continued Step 4 Find the similarity ratio. Since Holt Geometry , ABC A’B’C’ by SSS .

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 4 Graph the image of MNP after a dilation with scale factor 3. Verify that M'N'P' MNP. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 4 Continued Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of M’N’P’. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 4 Continued Step 2 Graph M’N’P’. 7 6 5 4 3 2 1 -7 -6 -5 -4 -3 -2 -1 -1 -2 -3 -4 -5 -6 -7 Holt Geometry 1 2 3 4 5 6 7 X Y

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 4 Continued Step 3 Use the Distance Formula to find the side lengths. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Check It Out! Example 4 Continued Step 4 Find the similarity ratio. Since Holt Geometry , MNP M’N’P’ by SSS .

Dilations and Similarity 7-6 in the Coordinate Plane Lesson Quiz: Part I 1. Given X(0, 2), Y(–2, 2), and Z(–2, 0), find the coordinates of X', Y, and Z' after a dilation with scale factor –4. X'(0, –8); Y'(8, –8); Z'(8, 0) 2. JOK LOM. Find the coordinates of M and the scale factor. Holt Geometry

Dilations and Similarity 7-6 in the Coordinate Plane Lesson Quiz: Part II 3. Given: A(–1, 0), B(–4, 5), C(2, 2), D(2, –1), E(–4, 9), and F(8, 3) Prove: ABC DEF Therefore, by SSS . Holt Geometry and ABC DEF