# 9.2: Investigating Special Triangles

**At Grade**Created by: CK-12

*This activity is intended to supplement Geometry, Chapter 8, Lesson 4.*

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Problem 1 – Investigation of 45∘−45∘−90∘ Triangles

First, turn on your TI-84 and press **APPS**. Arrow down until you see **Cabri Jr** and press **ENTER**. Open the file **ISOSC**. This file has a triangle with an isosceles triangle with

Using the **Perpendicular** tool (**ZOOM** > **Perp.**), construct a perpendicular from point **Alph-Num** tool (**GRAPH** > **Alph-Num**), select the point, and press **ENTER** for the letter

Construct line segments **Segment**) and then measure the segments (**GRAPH** > **Measure** > **D. & Length**).

Would you have expected these segments to be equal in length?

Drag point

Will they always be equal? _____________________________

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Problem 2 – Investigation of 30∘−60∘−90∘ Triangles

Open the file **EQUIL**. Note that all three angles are

Construct the perpendicular from

From the construction above, we know that

Construct segment

This completes the construction of two

Measure the three sides of triangle

Press **Calculate** tool. Click on the length of

Drag point

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Problem 3 – Investigation of 45∘−45∘−90∘ Triangles

Press the o button and select **New** to open a new document.

To begin the construction of the

Use the compass tool with center \begin{align*}A\end{align*} and radius \begin{align*}AB\end{align*}. The circle will intersect the perpendicular line at \begin{align*}C\end{align*}.

Hide the circle and construct segments \begin{align*}AC\end{align*} and \begin{align*}BC\end{align*}.

Explain why \begin{align*}AB = AC\end{align*} and why angle \begin{align*}ACB = \text{angle} \ ABC\end{align*}?

Why are these two angles \begin{align*}45^\circ\end{align*} each?

Measure the sides of the triangle.

\begin{align*}AC = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && BC = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && AB = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\end{align*}

Use the **Calculate** tool to find the ratio of \begin{align*}AC:BC\end{align*} and \begin{align*}AC:AB\end{align*}. Once again, these ratios will be important when you study trigonometry.

Drag point \begin{align*}B\end{align*} and observe what happens to the sides and ratios.

Why do the ratios remain constant while the sides change?