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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 CAR\begin{align*}\triangle{CAR}\end{align*}. You are also given DS¯¯¯¯¯¯¯\begin{align*}\overline{DS}\end{align*} which is parallel to side CR\begin{align*}CR\end{align*}.

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

Position AD\begin{align*}AD\end{align*} DC\begin{align*}DC\end{align*} AS\begin{align*}AS\end{align*} SR\begin{align*}SR\end{align*} ADDC\begin{align*}\frac{AD}{DC}\end{align*} ASSR\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 ADDC\begin{align*}\frac{AD}{DC}\end{align*} and ASSR\begin{align*}\frac{AS}{SR}\end{align*}. If side DS\begin{align*}DS\end{align*} is parallel to side CR\begin{align*}CR\end{align*}, then _____________.

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

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

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

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

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

8. Find the value of x\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 U\begin{align*}U\end{align*} to 2 different positions and point N\begin{align*}N\end{align*} to 2 different positions and collect the data in the table on the accompanying worksheet.

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

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

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