8.2: Side-Splitter Theorem
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
- This activity was written to be explored with the Cabri Jr. app on the TI-84.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
- To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=12318 and select SIDESP1, SIDESP2, and SIDESP3.
Associated Materials
- Student Worksheet: Side-Splitter Theorem http://www.ck12.org/flexr/chapter/9692, scroll down to the second activity.
- Cabri Jr. Application
- SIDESP1.8xv, SIDESP2.8xv, and SIDESP3.8xv
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
1. Sample answers:
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*}
9. Sample answers:
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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