# 9.2: Investigating Special Triangles

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

## Problem 1 – Investigation of 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 to side . Label the point of intersection of this line with as . To name the point, they need to select the **Alph-Num** tool (**GRAPH** > **Alph-Num**), select the point, and press **ENTER** for the letter .

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

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

Drag point to see the effect on the lengths of the line segments. It appears that the perpendicular from the vertex always bisects the opposite side. Measure the angles and .

Will they always be equal? _____________________________

## Problem 2 – Investigation of Triangles

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

Construct the perpendicular from to side . Label the point of intersection as .

From the construction above, we know that bisects and that .

Construct segment . We now have triangle where , and . We also have triangle where , and .

This completes the construction of two triangles. We will work only with the triangle .

Measure the three sides of triangle .

Press and select the **Calculate** tool. Click on the length of , then on the length of . Press the key. Move it to the upper corner. Repeat this step to find the ratio of and . These ratios will become important when you start working with trigonometry.

Drag point to another location. What do you notice about the three ratios?

## Problem 3 – Investigation of Triangles

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

To begin the construction of the triangle, construct line segment and a perpendicular to at .

Use the compass tool with center and radius . The circle will intersect the perpendicular line at .

Hide the circle and construct segments and .

Explain why and why angle ?

Why are these two angles each?

Measure the sides of the triangle.

Use the **Calculate** tool to find the ratio of and . Once again, these ratios will be important when you study trigonometry.

Drag point and observe what happens to the sides and ratios.

Why do the ratios remain constant while the sides change?