8.1: Constructing Similar Triangles
This activity is intended to supplement Geometry, Chapter 7, Lesson 4.
ID: 8179
Time required: 30 minutes
Activity Overview
In this activity, students will investigate three different methods of constructing similar triangles. They will use the Dilation tool with the dilation point inside and outside of the triangle to investigate different relationships. Also, students will use the Parallel tool to construct two similar triangles from one triangle.
Topic: Ratio, Proportion & Similarity
- Use inductive reasoning to classify each set of conditions as necessary and/or sufficient for similarity:
a) the side lengths of one triangle are equal to the corresponding side lengths of another triangle.
b) two triangles of one triangle are congruent to two angles of another triangle.
Teacher Preparation
- This activity is designed to be used in a high school geometry classroom.
- Before starting this activity, students should be familiar with the term dilation.
- This activity is intended to be mainly teacher-led, with breaks for individual student work. Use the following pages to present the material to the class and encourage discussion. Students will follow along using their calculators.
- To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
Associated Materials
- Student Worksheet: Constructing Similar Triangles http://www.ck12.org/flexr/chapter/9692
- Cabri Jr. Application
Students need to press APPS and select CabriJr to start the application. When they open a new document, they need to make sure that the axes are hidden. If the axes are displayed, press GRAPH > Hide/Show > Axes.
Explain to students that similar triangles are those that have the same shape but not necessarily the same size.
Congruent triangles are a special type of similar triangle where corresponding sides are congruent. In similar triangles, corresponding angles are congruent but corresponding sides are proportional. In this activity, students will look at three methods of constructing similar triangles and will test these properties using dilations or stretches.
In order to examine all of the sides and angles, students should work in groups of three. Have one student (Student A on the worksheet) in each group construct the first triangle and save it as “SIMTRI.” This is saved in the TI-84 Plus family as an APPVAR. Student A needs to transfer the APPVAR to the other two students in the group.
Problem 1 – Similar Triangles using Dilation
Student A will use the Triangle tool to construct a triangle of any shape or size. Then they need to use the Alph-Num tool to label the vertices \begin{align*}P\end{align*}, \begin{align*}Q\end{align*}, and \begin{align*}R\end{align*}.
Student A should measure angle \begin{align*}P\end{align*} and side \begin{align*}PQ\end{align*}, Student B should measure angle \begin{align*}Q\end{align*} and side \begin{align*}QR\end{align*}, Student C should measure angle \begin{align*}R\end{align*} and side \begin{align*}PR\end{align*}.
Note: To increase the number of digits for the length of the side, hover the cursor over the measurement and press +.
With the Point tool, students on their own calculators will construct a point \begin{align*}C\end{align*} in the center of the triangle. They will then use the Alph-Num tool to place the number 2 at the top of the screen.
Explain to students that the point \begin{align*}C\end{align*} will be the center of the dilation and the number \begin{align*}2\end{align*} will be the scale factor.
With the Dilation tool selected, students need to move the cursor to point \begin{align*}C\end{align*}, the center of the dilation, and press ENTER. They should notice that the shape of the cursor changes, which is to indicate that everything will be stretched away from this point. Then students move the cursor to the perimeter of the triangle and press ENTER. Finally, move to the scale factor, 2, and press ENTER.
Students will see that a new, larger triangle will appear outside of \begin{align*}\triangle{PQR}\end{align*}. Have a prelimenary discussion with students about whether they think the new triangle is similar to \begin{align*}\triangle{PQR}\end{align*} and how they can confrim their hypothesis.
Instruct students to label this triangle as \begin{align*}XYZ\end{align*} so that \begin{align*}X\end{align*} corresponds to \begin{align*}P\end{align*}, \begin{align*}Y\end{align*} to \begin{align*}Q\end{align*} and \begin{align*}Z\end{align*} to \begin{align*}R\end{align*}. Each person in the group should select and measure their appropriate angle and side in the new triangle. Students should answer Questions 1 and 2 on the worksheet comparing the two angles and two sides.
Explain to students that \begin{align*}XY = 2\ PQ\end{align*} indicates that the sides have the ratio 2:1. If all three sides display the same result, then the sides are said to be proportional.
To answer Questions 3 and 4 on the worksheet, students will observe the changes in the triangles as they drag a point in the original triangle and the point of dilation.
Problem 2 – Different Scale Factors
Students will continue using the same file. To change the scale factor, students must select the Alph-Num tool, move the cursor to the scale factor of 2 and press ENTER. Then they should delete 2 by pressing DEL, press ALPHA to change the character to a number, and enter 3. Students can now answer Question 5 on the worksheet.
Now students will investigate another scale factor by changing it from 3 to 0.5. They should then answer Questions 6 and 7 on the worksheet, summarizing their observations of the triangles and their measurements.
For the last investigation in this problem, students will look at the effect of a dilation when the center point \begin{align*}C\end{align*} is outside of the pre-image triangle and a negative scale factor is used. Students will need to move \begin{align*}\triangle{PQR}\end{align*} by moving the cursor to one side and when all sides of the triangle flash, press ALPHA and then use the arrow keys.
Problem 3 – Similar Triangles with a Parallel Line
Finally, students will look at a completely different method of constructing similar triangles. All students will need to open a new file. It is the teacher’s decision to have them save the file. Student A should construct a triangle \begin{align*}PQR\end{align*} and transfer it to the others in their group. They will all measure their same side and angle as before.
All students need to construct a point on \begin{align*}\overline{PQ}\end{align*} using the Point on tool and label it \begin{align*}S\end{align*} using the Alph-Num tool. Using this point, they construct a line that is parallel to \begin{align*}\overline{QR}\end{align*} by selecting the Parallel tool, and then choosing the point and side.
With the Intersection tool, students need to find the intersection of side \begin{align*}PR\end{align*} and the parallel line, and then use the Alph-Num tool to label it as \begin{align*}T\end{align*}. The Hide/Show > Object tool enables students to hide the parallel line, and then construct segment \begin{align*}ST\end{align*} with the Segment tool. If students can prove that all of the angles are congruent, they have completed their first step in proving that the two triangles are similar.
With the Measure > D. & Length tool, students need to measure segments of the corresponding sides. In the screen to the right all three pairs of sides have been measured.
To complete the proof, students will need to confirm that all of the sides are proportional. Using the Calculate tool, they can select a side measurement, press the operation key \begin{align*}( \div )\end{align*} and then select the corresponding side measurement.
If all three ratios are equivalent, then the sides are proportional and the two triangles are similar.
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