Scale Factors

New York State Common Core Math Geometry, Module 2, Lesson 5

Worksheets for Geometry

In the last lesson, students learned about the triangle side splitter theorem, which is now used to prove the dilation theorem. In Grade 8 students learned about the fundamental theorem of similarity (FTS), which contains the concepts that are in the dilation theorem presented in this lesson. We call it the dilation theorem at this point in the module because students have not yet entered into the formal study of similarity. Some students may recall FTS from Grade 8 as they enter into the discussion following the Opening Exercise. Their prior knowledge of this topic will strengthen as they prove the dilation theorem.

Scale Factors

Classwork

Opening Exercise

Quick Write: Describe how a figure is transformed under a dilation with a scale factor = 1, 𝑟 > 1, and 0 < 𝑟 < 1

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Discussion

DILATION THEOREM: If a dilation with center 𝑂 and scale factor 𝑟 sends point 𝑃 to 𝑃′ and 𝑄 to 𝑄′, then |𝑃′𝑄′| = 𝑟|𝑃𝑄|. Furthermore, if 𝑟 ≠ 1 and 𝑂, 𝑃, and 𝑄 are the vertices of a triangle, then 𝑃𝑄 || 𝑃′𝑄′. Now consider the dilation theorem when 𝑂, 𝑃, and 𝑄 are the vertices of △ 𝑂𝑃𝑄. Since 𝑃′ and 𝑄′ come from a dilation with scale factor 𝑟 and center 𝑂, we have 𝑂𝑃′/𝑂𝑃 = 𝑂𝑄′/𝑂𝑄 = 𝑟.

There are two cases that arise; recall what you wrote in your Quick Write. We must consider the case when 𝑟 > 1 and when 0 < 𝑟 < 1. Let’s begin with the latter.

Exercises

1. Prove Case 2: If 𝑂, 𝑃, and 𝑄 are the vertices of a triangle and 𝑟 > 1, show that (a) 𝑃𝑄 || 𝑃′𝑄′ and (b) 𝑃′𝑄′ = 𝑟𝑃𝑄.
   Use the diagram below when writing your proof.
2. a. Produce a scale drawing of △ 𝐿𝑀𝑁 using either the ratio or parallel method with point 𝑀 as the center and a scale factor of 3/2.
   b. Use the dilation theorem to predict the length of 𝐿′𝑁′, and then measure its length directly using a ruler.
   c. Does the dilation theorem appear to hold true?
3. Produce a scale drawing of △ 𝑋𝑌𝑍 with point 𝑋 as the center and a scale factor of 1/4. Use the dilation theorem to predict 𝑌′𝑍′, and then measure its length directly using a ruler. Does the dilation theorem appear to hold true?
4. Given the diagram below, determine if △ 𝐷𝐸𝐹 is a scale drawing of △ 𝐷𝐺𝐻. Explain why or why not.

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