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# 9.2: Investigating Special Triangles

Difficulty Level: At Grade Created by: CK-12

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

## Problem 1 – Investigation of $45^\circ-45^\circ-90^\circ$ 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 $AB = AC$.

Using the Perpendicular tool (ZOOM > Perp.), construct a perpendicular from point $A$ to side $BC$. Label the point of intersection of this line with $BC$ as $D$. To name the point, they need to select the Alph-Num tool (GRAPH > Alph-Num), select the point, and press $x^{-1}$ ENTER for the letter $D$.

Construct line segments $BD$ and $CD$ ($\pi$ > Segment) and then measure the segments (GRAPH > Measure > D. & Length).

$BD = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && CD = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

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

Drag point $C$ 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 $BAD$ and $CAD$.

$\angle{BAD} = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && \angle{CAD} = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

Will they always be equal? _____________________________

## Problem 2 – Investigation of $30^\circ-60^\circ-90^\circ$ Triangles

Open the file EQUIL. Note that all three angles are $60^\circ$ angles.

Construct the perpendicular from $A$ to side $BC$. Label the point of intersection as $D$.

From the construction above, we know that $D$ bisects $BC$ and that $m\angle{BAD} = 30^\circ$.

Construct segment $BD$. We now have triangle $BAD$ where $m\angle{D} = 90^\circ$, $m\angle{B} = 60^\circ$ and $m\angle{A} = 30^\circ$. We also have triangle $ACD$ where $m\angle{A} = 30^\circ$, $m\angle{C} = 60^\circ$ and $m\angle{D} = 90^\circ$.

This completes the construction of two $30^\circ-60^\circ-90^\circ$ triangles. We will work only with the triangle $BAD$.

Measure the three sides of triangle $BAD$.

$AB = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && BD = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && AD = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

Press $\sigma$ and select the Calculate tool. Click on the length of $BD$, then on the length of $AB$. Press the $\infty$ key. Move it to the upper corner. Repeat this step to find the ratio of $AD:AB$ and $AD:BD$. These ratios will become important when you start working with trigonometry.

$BD:AB = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && AD:AB = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && AD:BD = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

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

## Problem 3 – Investigation of $45^\circ-45^\circ-90^\circ$ Triangles

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

To begin the construction of the $45^\circ-45^\circ-90^\circ$ triangle, construct line segment $AB$ and a perpendicular to $AB$ at $A$.

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

Hide the circle and construct segments $AC$ and $BC$.

Explain why $AB = AC$ and why angle $ACB = \text{angle} \ ABC$?

Why are these two angles $45^\circ$ each?

Measure the sides of the triangle.

$AC = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && BC = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} && AB = \underline{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}$

Use the Calculate tool to find the ratio of $AC:BC$ and $AC:AB$. Once again, these ratios will be important when you study trigonometry.

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

Why do the ratios remain constant while the sides change?

Feb 23, 2012

Nov 03, 2014