Dilations 6-7 Warm Up Lesson Presentation Lesson Quiz Holt McDougal Geometry Holt Geometry

Warm Up 1. Translate the triangle with vertices A(2, –1), B(4, 3), and C(–5, 4) along the vector <2, 2>. A'(4,1), B'(6, 5),C(–3, 6) 2. ∆ABC ~ ∆JKL. Find the value of JK.

Objective Identify and draw dilations.

Vocabulary center of dilation enlargement reduction

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 A’

Example 1: Identifying Dilations Tell whether each transformation appears to be a dilation. Explain. A. B. No; the figures are not similar. Yes; the figures are similar and the image is not turned or flipped.

Check It Out! Example 1 Tell whether each transformation appears to be a dilation. Explain. a. b. Yes, the figures are similar and the image is not turned or flipped. No, the figures are not similar.

K = Center of dilation CP PP’ PQ CQ QQ’ P’Q’ A dilation, or similarity transformation, is a transformation in which every point P and its image P’ have the same ratio.

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). k > 1 is an enlargement, or expansion. 0< k < 1 is a reduction, or contraction.

Example 2: Drawing Dilations Copy the figure and the center of dilation P. Draw the image of ∆WXYZ under a dilation with a scale factor of 2. Step 1 Draw a line through P and each vertex. Step 2 On each line, mark twice the distance from P to the vertex. W’ X’ Step 3 Connect the vertices of the image. Y’ Z’

Step 1 Draw a line through Q and each vertex. Check It Out! Example 2 Copy the figure and the center of dilation. Draw the dilation of RSTU using center Q and a scale factor of 3. Step 1 Draw a line through Q and each vertex. R’ S’ Step 2 On each line, mark twice the distance from Q to the vertex. Step 3 Connect the vertices of the image. T’ U’

Example 1: Drawing and Describing Dilations A. Apply the dilation D to the polygon with the given vertices. Describe the dilation. D: (x, y) → (3x, 3y) A(1, 1), B(3, 1), C(3, 2) A’ (3, 3), B’ (9, 3), C’ (9,6) scale factor 3

Example 1: Continued B. Apply the dilation D to the polygon with the given vertices. Describe the dilation. D: (x, y) → P(–8, 4), Q(–4, 8), R(4, 4) 4 3 x, y P’(-6, 3), Q’ (-3, 6), R’ (3, 3) scale factor 3/4

Check It Out! Example 1 Name the coordinates of the image points. Describe the dilation. (x, y)→ ( ¼ x, ¼ y) D(-8, 0), E(-8, -4), and F(-4, -8). D'(-2, 0), E'(-2, -1), F'(-1, -2); scale factor 1/4

Example 3: Drawing Dilations On a sketch of a flower, 4 in. represent 1 in. on the actual flower. If the flower has a 3 in. diameter in the sketch, find the diameter of the actual flower. The scale factor in the dilation is 4, so a 1 in. by 1 in. square of the actual flower is represented by a 4 in. by 4 in. square on the sketch. Let the actual diameter of the flower be d in. 3 = 4d d = 0.75 in.

Check It Out! Example 3 What if…? An artist is creating a large painting from a photograph into square and dilating each square by a factor of 4. Suppose the photograph is a square with sides of length 10 in. Find the area of the painting. The scale factor of the dilation is 4, so a 10 in. by 10 in. square on the photograph represents a 40 in. by 40 in. square on the painting. Find the area of the painting. A = l  w = 4(10)  4(10) = 40  40 = 1600 in2

If the scale factor of a dilation is negative, the preimage is rotated by 180°. For k > 0, a dilation with a scale factor of –k is equivalent to the composition of a dilation with a scale factor of k that is rotated 180° about the center of dilation.

Example 4: Drawing Dilations in the Coordinate Plane Draw the image of the triangle with vertices P(–4, 4), Q(–2, –2), and R(4, 0) under a dilation with a scale factor of centered at the origin. The dilation of (x, y) is

Graph the preimage and image. Example 4 Continued Graph the preimage and image. P P’ Q’ R’ R Q

Check It Out! Example 4 Draw the image of the triangle with vertices R(0, 0), S(4, 0), T(2, -2), and U(–2, –2) under a dilation centered at the origin with a scale factor of . The dilation of (x, y) is

Check It Out! Example 4 Continued Graph the preimage and image. R’ S’ T’ U’ R S T U

Determine whether the polygons are similar. A(–6, -6), B(-6, 3), C(3, 3), D(3, -6) H(-2, -2), J(-2, 1), K(1, 1), L(1, -2) ABCD maps to HJKL (x, y) → 1 3 x y ,

Determine whether the polygons are similar. P(2, 0), Q(2, 4), R(4, 4),S(4, 0) W(5, 0), X(5, 10), Y(8, 10), Z(8, 0). No; (x, y) → (2.5x, 2.5y) maps P to W, but not S to Z.

Determine whether the polygons are similar. Yes; translation: A(1, 2), B(2, 2), C(1, 4) D(4, -6), E(6, -6), F(4, -2) Yes; translation: (x, y) → (x + 1, y - 5). dilation: (x, y) → (2x, 2y).

Determine whether the polygons are similar. F(3, 3), G(3, 6), H(9, 3), J(9, –3) S(–1, 1), T(–1, 2), U(–3, 1), V(–3, –1) Yes; reflection: (x, y) → (-x, y). dilation: (x, y) → (1/3 x, 1/3 y)

Determine whether the polygons are similar. Yes; rotation: A(2, -1), B(3, -1), C(3, -4) P(3, 6), Q(3, 9), R(12, 9). Yes; rotation: (x, y) → (-y, x) dilation: (x, y) → (3x, 3y)

Determine whether the polygons are similar Determine whether the polygons are similar. If so, describe the transformation in 2 different ways, from the larger to the smaller, and the smaller to the larger.

Determine whether the polygons are similar Determine whether the polygons are similar. If so, describe the transformation in 2 different ways, from the larger to the smaller, and the smaller to the larger.

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.

Example 3 Continued Step 2 Use the Distance Formula to find the side lengths.

Example 3 Continued Step 3 Find the similarity ratio. = 2 = 2 Since and E  E, by the Reflexive Property, ∆EHJ ~ ∆EFG by SAS ~ .

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

Check It Out! Example 3 Continued Step 1 Plot the points and draw the triangles. R S T U V

Check It Out! Example 3 Continued Step 2 Use the Distance Formula to find the side lengths.

Check It Out! Example 3 Continued Step 3 Find the similarity ratio. Since and R  R, by the Reflexive Property, ∆RST ~ ∆RUV by SAS ~ .

Example 4: Using the SSS Similarity Theorem Graph the image of ∆ABC after a dilation with scale factor Verify that ∆A'B'C' ~ ∆ABC.

Example 4 Continued Step 1 Multiply each coordinate by to find the coordinates of the vertices of ∆A’B’C’.

Example 4 Continued Step 2 Graph ∆A’B’C’. B’ (2, 4) A’ (0, 2)

Example 4 Continued Step 3 Use the Distance Formula to find the side lengths.

Example 4 Continued Step 4 Find the similarity ratio. Since , ∆ABC ~ ∆A’B’C’ by SSS ~.

Check It Out! Example 4 Graph the image of ∆MNP after a dilation with scale factor 3. Verify that ∆M'N'P' ~ ∆MNP.

Check It Out! Example 4 Continued Step 1 Multiply each coordinate by 3 to find the coordinates of the vertices of ∆M’N’P’.

Check It Out! Example 4 Continued Step 2 Graph ∆M’N’P’.

Check It Out! Example 4 Continued Step 3 Use the Distance Formula to find the side lengths.

Check It Out! Example 4 Continued Step 4 Find the similarity ratio. Since , ∆MNP ~ ∆M’N’P’ by SSS ~.

Lesson Quiz: Part I 1. Tell whether the transformation appears to be a dilation. yes 2. Copy ∆RST and the center of dilation. Draw the image of ∆RST under a dilation with a scale of .

Lesson Quiz : Part-I 3. Apply the dilation D: (x, y) to the polygon with vertices A(2, 4), B(2, 6), and C(6, 4). Name the coordinates of the image points. Describe the dilation. A’(3, 6), B’(3, 9), C’(9, 6); k= 3/2 Given X(0, 2), Y(–2, 2), and Z(–2, 0), find the dilation with scale factor –4. X'(0, –8); Y'(8, –8); Z'(8, 0)

Lesson Quiz : Part-II Determine whether the polygons with the given vertices are similar. 5. A(-4, 4), B(6, 4), C(6, -4), D(-4, -4) and P(-2, 2), Q(4, 2), R(4, -2), S(-2, -2) No; (x, y) → (0.5x, 0.5y) maps A to P, but not B to Q. 6. A(2, 2), B(2, 4), C(6, 4) and D(3, -3), E(3, -6), F(9, -6) Yes; △ ABC maps to △ A’B’C’ by a reflection: (x, y) → (x, -y). Then △ A’B’C’ maps to △DEF by a dilation:(x, y) → (1.5x, 1.5y).

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, and ∆ABC ~ ∆DEF by SSS ~.