Lesson 3.4 Core Focus on Geometry Dilations

Warm-Up Determine the type of transformation occurring in each graph. Explain your reasoning. 1. 2. 3. Translation; figure moved up and left Reflection; figure “flipped” over the x-axis Rotation; figure turned around the origin

Dilate an image on a coordinate plane. Lesson 3.4 Dilations Dilate an image on a coordinate plane.

Vocabulary Dilation A transformation that changes the size of a figure, but does not change the shape. Good to Know!  The image of a dilation is either an enlarged or reduced version of the pre-image.  Similar to rotations, dilations have a point which serves as the center for dilation. Scale Factor The ratio of two corresponding sides of the pre-image and image.

Dilations on a Coordinate Plane A figure is dilated using scale factor b when each x- and y-coordinate is multiplied by b (b ≠ 0). The dilation transformation rule for scale factor b is: (x, y) → (bx, by)

Example 1 A B C ∆ABC is drawn at right. Find its image under a dilation centered at the origin with a scale factor of 2. Graph the image. Multiply the coordinates of each point by the scale factor of 2. A(−2, 2)  (2  −2, 2  2)  A(−4, 4) B(1, −2)  (2  1, 2  −2)  B(2, −4) C(−2, −2)  (2  −2, 2  −2)  C(−4, −4) Graph the image. Notice that each side length of the image is twice as long as the side lengths on the pre-image.

Explore! Bigger Or Smaller? A dilation that creates an image larger than the pre-image is called an enlargement. A dilation that creates an image smaller than the pre-image is a reduction. Certain scale factor values create enlarged images compared to the pre-image. Other values create images that are reduced in size. In this Explore!, you will determine which values of the scale factor create each of these situations. Step 1 Graph H(2, 2), J(4, 2) and K(4, 4) on a coordinate plane. Step 2 Using a colored pencil, graph the image ΔH’J’K’ after a dilation with a scale factor of 0.5 on the same coordinate plane. Step 3 Using a different colored pencil, graph the image ΔHJK after a dilation with a scale factor of 2 on the same coordinate plane. Step 4 If b represents the scale factor, write an inequality that describes scale factors that create enlargements. Write another inequality that describes scale factors that create reductions. How did you arrive at your answer? Step 5 What type of image does a scale factor of 1 create? Support your answer with evidence.

Types Of Dilations (x, y) → (bx, by) When the absolute value of the scale factor, b, is greater than 1, the image created by the dilation is an enlargement. When the absolute value of the scale factor, b, is between 0 and 1, the image created by the dilation is a reduction.

Example 2 Give the scale factor for each dilation. Determine if the scale factor creates an enlargement or a reduction in size compared to the pre-image. a. (x, y) → (0.3x, 0.3y) The scale factor is the value that each coordinate is multiplied by. The scale factor is 0.3. Scale Factor = 0.3 Compare the scale factor to 1. 0.3 < 1 Since the scale factor is less than 1, the dilation is a reduction.

Example 2 Continued… Give the scale factor for each dilation. Determine if the scale factor creates an enlargement or a reduction in size compared to the pre-image. b. (x, y) → The scale factor is the value that each coordinate is multiplied by. The scale factor is . Scale Factor = Compare the scale factor to 1. Since the scale factor is greater than 1, the dilation is an enlargement.

Communication Prompt How is a dilation similar to and/or different from the other types of transformations in this block?

Exit Problem Graph the pre-image of ΔSHP and its image under the dilation. S(9, −6), H(−3, −3) and P(0, 6) P P H H S S