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Understanding Scale Factors in Photo Stickers and Dilations

Learn how to determine the scale factor for photo stickers and dilations in geometry. This guide explains how to find the scale factor by using the ratio of the side lengths of the sticker image and the original photo. Through examples, we illustrate how to calculate the scale factor based on dimensions and check if dilations maintain consistent ratios. You will discover how the scale factor applies when manipulating coordinates and understand the significance of origin points in geometric transformations.

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Understanding Scale Factors in Photo Stickers and Dilations

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  1. Photo Stickers You are making your own photo stickers. Your photo is 4 inches by 4 inches. The image on the stickers is 1.1 inches by 1.1 inches. What is the scale factor of the reduction? EXAMPLE 3 Find a scale factor

  2. The scale factor is the ratio of a side length of the sticker image to a side length of the original photo, or . In simplest form, the scale factor is . 11 40 1.1 in. 4 in. EXAMPLE 3 Find a scale factor SOLUTION

  3. EXAMPLE 4 Standardized Test Practice SOLUTION Determine if EFGHis a dilation of PQRSby checking whether the same scale factor can be used to obtain E, F, and G from P, Q, and R.

  4. P(3, 0) E(9, 0) Q(1, 1) F(3, 3) R(0, 2) G(0, 6) S(4, 5) H(3 4, 3 5) = H(12, 15) ANSWER (x, y) (kx, ky) The correct answer is C. CHECK:Draw rays from the origin through each point and its image. EXAMPLE 4 Standardized Test Practice k = 3 k = 3 k = 3 Because k is the same in each case, the image is a dilation with a scale factor of 3. So, you can use the scale factor to find the image H of point S.

  5. The scale factor is the ratio of a side length of the sticker image to a side length of the original 1.1 in. photo, or . 5.5 ANSWER In simplest form, the scale factor is . 1 5 for Examples 3 and 4 GUIDED PRACTICE 3. What If? In Example 3, what is the scale factor of the reduction if your photo is 5.5 inches by 5.5 inches? SOLUTION

  6. ANSWER A dilation with respect to the origin and scale factor k can be described as (x, y) (kx ,ky) if (x,y) = (0,0) then (kx, ky) = (k 0, k 0) = (0, 0). for Examples 3 and 4 GUIDED PRACTICE 4. Suppose a figure containing the origin is dilated. Explain why the corresponding point in the image of the figure is also the origin.

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