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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.

ID: 12318

Time Required: 15 minutes

## Activity Overview

In this activity, students will explore the side-splitter theorem.

Topic: Ratio, Proportion & Similarity

• Side-Splitter Theorem

Teacher Preparation and Notes

Associated Materials

## Problem 1 – Side-Splitter Theorem

Students will begin this activity by looking at the side-splitter theorem. Students are given a triangle with a segment parallel to one side. They will discover that if a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Students will be asked to collect data by moving point \begin{align*}A\end{align*} and point \begin{align*}D\end{align*}. Students are asked several questions about the relationships in the triangle.

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

In Problem 2, students will be asked to apply the side-splitter theorem to several homework problems.

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

In Problem 3, students will discover the corollary to the Side-Splitter Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally.

Students are asked several questions about the corollary to the side-splitter theorem.

## Solutions

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*}
1 4.1 1.8 4.6 2.0 2.27 2.30
2 3 2.8 3.4 3.2 1.07 1.06
3 2.6 2.4 3.2 3.0 1.08 1.07
4 3.3 3.1 2.5 2.3 1.06 1.08

2. The ratios of the side lengths are equal.

3. \begin{align*}\frac{AD}{DC} = \frac{AS}{SR}\end{align*}

4. The ratio remains the same.

5. The ratio changes when moving point \begin{align*}D\end{align*}.

6. When you move the parallel line, you are changing the proportion between the upper and lower segments. When you move the point, the segments may get longer or shorter, but the proportion stays the same.

7. \begin{align*}12\end{align*}

8. \begin{align*}16.8\end{align*}

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*}
1 2.1 1.5 2.4 1.8 1.4 1.4
2 2.1 1.3 2.7 1.7 1.6 1.6
3 1.9 1.5 2.4 2.0 1.2 1.2
4 2.7 0.7 3.5 0.9 4 4

10. The ratios are equal.

11. \begin{align*}\frac{RN}{NO}\end{align*} and \begin{align*}\frac{EA}{AS}\end{align*} are congruent

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