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# 8.2: Side-Splitter Theorem

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 7, Lesson 5.

## Problem 1 – Side Splitter Theorem

In SIDESP1.8xv, you are given \begin{align*}\triangle{CAR}\end{align*}. You are also given \begin{align*}\overline{DS}\end{align*} which is parallel to side \begin{align*}CR\end{align*}.

1. Move point \begin{align*}D\end{align*} to 2 different positions and point \begin{align*}A\end{align*} to 2 different positions and collect the data in the table below. Calculate the ratios of \begin{align*}AD\end{align*} to \begin{align*}DC\end{align*} and \begin{align*}AS\end{align*} to \begin{align*}SR\end{align*} for each position and record the calculation in the table below.

Position \begin{align*}AD\end{align*} \begin{align*}DC\end{align*} \begin{align*}AS\end{align*} \begin{align*}SR\end{align*} \begin{align*}\frac{AD}{DC}\end{align*} \begin{align*}\frac{AS}{SR}\end{align*}
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2. Make some observations about the ratios of the sides in the triangle. What relationships do you notice?

3. Use the table to complete the following conjecture about the relationship between \begin{align*}\frac{AD}{DC}\end{align*} and \begin{align*}\frac{AS}{SR}\end{align*}. If side \begin{align*}DS\end{align*} is parallel to side \begin{align*}CR\end{align*}, then _____________.

4. In SIDESP2.8xv, drag point \begin{align*}A\end{align*}. Make some observations about the relationship of the ratios \begin{align*}\frac{AD}{DC}\end{align*} and \begin{align*}\frac{AS}{SR}\end{align*}?

5. In SIDESP2.8xv, drag point \begin{align*}D\end{align*}. Make some observations about the relationship of the ratios \begin{align*}\frac{AD}{DC}\end{align*} and \begin{align*}\frac{AS}{SR}\end{align*}?

6. Why are the results different when moving point \begin{align*}A\end{align*} versus moving point \begin{align*}D\end{align*}?

## Problem 2 – Application of the Side-Splitter Theorem

7. Find the value of \begin{align*}x\end{align*}.

8. Find the value of \begin{align*}x\end{align*}.

## Problem 3 – Extension of the Side-Splitter Theorem

For this problem, we will look at a corollary of the side-splitter theorem.

9. In SIDESP3.8xv, move point \begin{align*}U\end{align*} to 2 different positions and point \begin{align*}N\end{align*} to 2 different positions and collect the data in the table on the accompanying worksheet.

Position \begin{align*}RN\end{align*} \begin{align*}NO\end{align*} \begin{align*}EA\end{align*} \begin{align*}AS\end{align*} \begin{align*}\frac{RN}{NO}\end{align*} \begin{align*}\frac{EA}{AS}\end{align*}
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10. What do you notice about the ratios \begin{align*}\frac{RN}{NO}\end{align*} and \begin{align*}\frac{EA}{AS}\end{align*}?

11. Use the table to complete the following conjecture about the relationship between \begin{align*}\frac{RN}{NO}\end{align*} and \begin{align*}\frac{EA}{AS}\end{align*}. If lines \begin{align*}RE\end{align*}, \begin{align*}NA\end{align*}, and \begin{align*}OS\end{align*} are parallel and cut by two transversals, then ________________________.

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