12-7 Dilations Warm Up Lesson Presentation Lesson Quiz 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 the shape. The image and the preimage of a figure under a dilation are similar.

A dilation enlarges or reduces all dimensions proportionally A dilation enlarges or reduces all dimensions proportionally. A dilation with a scale factor greater than 1 is an enlargement, or expansion. A dilation with a scale factor greater than 0 but less than 1 (a fraction) is a reduction, or contraction.

Examples 1 and 2: Identifying Dilations Tell whether each transformation appears to be a dilation. Explain. 1. 2. No; the figures are not similar. Yes; the figures are similar and the image is not turned or flipped.

Examples 3 and 4: Identifying Dilations Tell whether each transformation appears to be a dilation. Explain. 3. 4. Yes, the figures are similar and the image is not turned or flipped. No, the figures are not similar.

For a dilation with scale factor k, if k > 0, the figure is not turned or flipped. If k < 0, the figure is rotated by 180°. Helpful Hint

Example 5: Drawing Dilations Draw the image of WXYZ under a dilation with a scale factor of 2 using the center of dilation P. Step 1 Draw a ray starting at P and pass through each vertex. Step 2 On each ray, mark twice the distance from P to the vertex. W’ X’ Z’ Y’ Step 3 Connect the vertices of the image.

Example 6: Drawing Dilations Draw the dilation of RSTU using center Q and a scale factor of 3. Step 1 Draw a ray starting at Q and passing through each vertex. R’ S’ Step 2 On each ray, mark twice the distance from Q to the vertex. Step 3 Connect the vertices of the image. T’ U’

Example 7: Real-Life Dilations What if…? An artist is creating a large painting from a photograph divided into squares 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 8: 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 8 Continued Graph the preimage and image. P P’ Q’ R’ R Q

Example 9 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

Graph the preimage and image. Example 9 Continued Graph the preimage and image. R’ S’ T’ U’ R S T U

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 II 3. A rectangle on a transparency has length 6cm and width 4 cm and with 4 cm. On the transparency 1 cm represents 12 cm on the projection. Find the perimeter of the rectangle in the projection. 240 cm 4. Draw the image of the triangle with vertices E(2, 1), F(1, 2), and G(–2, 2) under a dilation with a scale factor of –2 centered at the origin.