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