5Recall that 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.
6Example 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.
7Check It Out! Example 1Tell 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.
8For 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
10A 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 is a reduction, or contraction.
11Example 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’
12Step 1 Draw a line through Q and each vertex. Check It Out! Example 2Copy 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’
13Example 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 = 4dd = 0.75 in.
14Check It Out! Example 3What 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
16If 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.
17Example 4: Drawing Dilations in the Coordinate Plane Draw the image of the triangle with verticesP(–4, 4), Q(–2, –2), and R(4, 0) under adilation with a scale factor of centered at the origin.The dilation of (x, y) is
18Graph the preimage and image. Example 4 ContinuedGraph the preimage and image.PP’Q’R’RQ
19Check It Out! Example 4Draw 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 ofThe dilation of (x, y) is
20Check It Out! Example 4 Continued Graph the preimage and image.R’S’T’U’RSTU
21Lesson Quiz: Part I1. Tell whether the transformation appears to be a dilation.yes2. Copy ∆RST and the center of dilation. Draw the image of ∆RST under a dilation with a scale of .
22Lesson Quiz: Part II3. 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 cm4. Draw the image of the triangle with verticesE(2, 1), F(1, 2), and G(–2, 2) under a dilation with a scale factor of –2 centered at the origin.