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# 8.1: Construct Similar Triangles

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Geometry, Chapter 7, Lesson 4.

## Problem 1 – Similar Triangles using Dilation

• Open Cabri Jr. and open a new file.

Student A: Construct a triangle and label the vertices P\begin{align*}P\end{align*}, Q\begin{align*}Q\end{align*}, and R\begin{align*}R\end{align*}. Send the file to Students B and C. Measure P\begin{align*}\angle{P}\end{align*} and PQ¯¯¯¯¯¯¯¯\begin{align*}\overline{PQ}\end{align*}.

Student B: Measure Q\begin{align*}\angle{Q}\end{align*} and QR¯¯¯¯¯¯¯¯\begin{align*}\overline{QR}\end{align*}.

Student C: Measure R\begin{align*}\angle{R}\end{align*} and PR¯¯¯¯¯¯¯¯\begin{align*}\overline{PR}\end{align*}.

Note: Place the measurements in the top right corner.

• Construct point C\begin{align*}C\end{align*} in the center of the triangle.
• Place the number 2 on the screen.
• Select the Dilation tool and then select point C\begin{align*}C\end{align*}, the triangle, and the number 2.
• Label the triangle that appears, XYZ\begin{align*}XYZ\end{align*}, so that X\begin{align*}X\end{align*} corresponds to P\begin{align*}P\end{align*}, Y\begin{align*}Y\end{align*} to Q\begin{align*}Q\end{align*} and Z\begin{align*}Z\end{align*} to R\begin{align*}R\end{align*}.

Student A: Measure X\begin{align*}\angle{X}\end{align*} and XY¯¯¯¯¯¯¯¯\begin{align*}\overline{XY}\end{align*}.

Student B: Measure Y\begin{align*}\angle{Y}\end{align*} and YZ¯¯¯¯¯¯¯\begin{align*}\overline{YZ}\end{align*}.

Student C: Measure Z\begin{align*}\angle{Z}\end{align*} and XZ¯¯¯¯¯¯¯¯\begin{align*}\overline{XZ}\end{align*}.

1. What do you notice about the two angles? Compare this to the other students in your group.

2. How do the lengths of the sides compare? Is this the result that you were expecting?

3. Drag your point in PQR\begin{align*}\triangle{PQR}\end{align*}. Do the corresponding angles remain congruent? Does the relationship between corresponding sides remain the same? Compare your results to others in your group.

4. Drag point C\begin{align*}C\end{align*}. Are the relationships preserved under this change? Compare your results to others in your group. Does it make any difference that each person may have constructed a different center point.

## Problem 2 – Different Scale Factors

Using the Alph-Num tool, change the scale factor from 2 to 3.

5. What happens to your construction? Does this change the relationships you found in Problem 1?

6. Change the scale factor from 3 to 0.5. How does this affect your construction?

7. Summarize your findings by stating the effect of a dilation on corresponding angles and sides.

8. Drag PQR\begin{align*}\triangle{PQR}\end{align*} to the lower left corner and drag point C\begin{align*}C\end{align*} to the right of the triangle. Change the scale factor to -2. Are the properties that you noted above preserved by these changes?

## Problem 3 – Similar Triangles with a Parallel Line

Student A: Open a new file and construct a triangle PQR\begin{align*}PQR\end{align*}. Send the file to Students B and C. Measure P\begin{align*}\angle{P}\end{align*} and PQ¯¯¯¯¯¯¯¯\begin{align*}\overline{PQ}\end{align*}.

Student B: Measure Q\begin{align*}\angle{Q}\end{align*} and QR¯¯¯¯¯¯¯¯\begin{align*}\overline{QR}\end{align*}.

Student C: Measure R\begin{align*}\angle{R}\end{align*} and PR¯¯¯¯¯¯¯¯\begin{align*}\overline{PR}\end{align*}.

• Construct a point on PQ¯¯¯¯¯¯¯¯\begin{align*}\overline{PQ}\end{align*} and label it S\begin{align*}S\end{align*}.
• Construct a line through S\begin{align*}S\end{align*} that is parallel to QR¯¯¯¯¯¯¯¯\begin{align*}\overline{QR}\end{align*}.
• Label the point of intersection of side PR\begin{align*}PR\end{align*} and the parallel line as T\begin{align*}T\end{align*}.
• Hide the parallel line and construct line segment ST\begin{align*}ST\end{align*}.

9. Can you prove that all three pairs of corresponding angles are congruent? If so, then PST\begin{align*}\triangle{PST}\end{align*} is similar to PQR\begin{align*}\triangle{PQR}\end{align*}.

10. Calculate the ratio of PQ:PS\begin{align*}PQ:PS\end{align*}. Then calculate the ratios of the other sides. If all the ratios are equivalent, then the sides are proportional. Are the sides in PST\begin{align*}\triangle{PST}\end{align*} and PQR\begin{align*}\triangle{PQR}\end{align*} proportional?

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