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Common Core For Geometry

Worksheets for Geometry, Module 2, Lesson 5

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, π > 1, and 0 < π < 1

**Discussion**

**DILATION THEOREM**: If a dilation with center π and scale factor π sends point π to πβ² and π to πβ², then |πβ²πβ²| = π|ππ|.
Furthermore, if π β 1 and π, π, and π are the vertices of a triangle, then ππ || πβ²πβ².
Now consider the dilation theorem when π, π, and π are the vertices of β³ πππ. Since πβ² and πβ² come from a dilation
with scale factor π and center π, we have
ππβ²/ππ = ππβ²/ππ = π.

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

**Exercises**

- Prove Case 2: If π, π, and π are the vertices of a triangle and π > 1, show that (a) ππ || πβ²πβ² and (b) πβ²πβ² = πππ.

Use the diagram below when writing your proof. - a. Produce a scale drawing of β³ πΏππ using either the ratio or parallel method with point π as the center and a
scale factor of 3/2.

b. Use the dilation theorem to predict the length of πΏβ²πβ², and then measure its length directly using a ruler.

c. Does the dilation theorem appear to hold true? - Produce a scale drawing of β³ πππ with point π as the center and a scale factor of 1/4. Use the dilation theorem to predict πβ²πβ², and then measure its length directly using a ruler. Does the dilation theorem appear to hold true?
- Given the diagram below, determine if β³ π·πΈπΉ is a scale drawing of β³ π·πΊπ». Explain why or why not.

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