9.2: Investigating Special Triangles
This activity is intended to supplement Geometry, Chapter 8, Lesson 4.
ID: 7896
Time required: 45 minutes
Activity Overview
In this activity, students will investigate the properties of an isosceles triangle. Then students will construct a \begin{align*}30^\circ  60^\circ90^\circ\end{align*}
Topic: Right Triangles & Trigonometric Ratios

Calculate the trigonometric ratios for \begin{align*}45^\circ45^\circ  90^\circ, 60^\circ  60^\circ  60^\circ\end{align*}
45∘−45∘−90∘,60∘−60∘−60∘ and \begin{align*}30^\circ60^\circ90^\circ\end{align*}30∘−60∘−90∘ triangles.
Teacher Preparation and Notes
This activity is designed to be used in a high school or middle school geometry classroom.
 If needed, review or introduce the term median of a triangle. Any median of an equilateral triangle is also an altitude, angle bisector, and perpendicular bisector.
 This activity is designed to be studentcentered with the teacher acting as a facilitator while students work cooperatively.
 The worksheet guides students through the main ideas of the activity and provides a place for students to record their work. You may wish to have the class record their answers on separate sheets of paper, or just use the questions posed to engage a class discussion.
 To download Cabri Jr, go to http://www.education.ti.com/calculators/downloads/US/Software/Detail?id=258#.
 To download the calculator files, go to http://www.education.ti.com/calculators/downloads/US/Activities/Detail?id=7896 and select EQUIL and ISOSC.
Associated Materials
 Student Worksheet: Investigating Special Right Triangles http://www.ck12.org/flexr/chapter/9693, scroll down to the second activity.
 Cabri Jr. Application
 EQUIL.8xv and ISOSC.8xv
Problem 1 – Investigation of Triangles
First, turn on your TI84 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 \begin{align*}AB = AC\end{align*}
Using the Perpendicular tool (ZOOM > Perp.), construct a perpendicular from point \begin{align*}A\end{align*}
Construct line segments \begin{align*}BD\end{align*}
Would you have expected these segments to be equal in length?
Drag point \begin{align*}C\end{align*}
Will they always be equal?
Problem 2 – Investigation of 306090 Triangles
Open the file EQUIL. Note that all three angles are \begin{align*}60^\circ\end{align*}
Construct the perpendicular from \begin{align*}A\end{align*}
From the construction above, we know that \begin{align*}D\end{align*}
Construct segment \begin{align*}BD\end{align*}
This completes the construction of two \begin{align*}30^\circ  60^\circ  90^\circ\end{align*} triangles. We will work only with the triangle \begin{align*}BAD\end{align*}.
You may choose to have the students hide the segments \begin{align*}AC\end{align*} and \begin{align*}CD\end{align*}. To do this, construct segments \begin{align*}BD\end{align*} and \begin{align*}AB\end{align*} on top of the larger triangle. Then hide the original triangle. Keep the point \begin{align*}C\end{align*}. We will need that point later to resize the triangle.
Measure the three sides of the triangle.
Press GRAPH and select the Calculate tool. Click on the length of \begin{align*}BD\end{align*}, then on the length of \begin{align*}AB\end{align*}. Press the \begin{align*}\div\end{align*} key. Students will see the result 0.5. Move it to the upper corner. Repeat this step to find the ratio of \begin{align*}AD:AB\end{align*} and \begin{align*}AD:BD\end{align*}. These ratios will become important when you start working with trigonometry.
Drag point \begin{align*}C\end{align*} to another location.
What do you notice about the three ratios?
Problem 3 – Investigation of 454590 Triangles
Press the \begin{align*}Y=\end{align*} button and select New to open a new document.
To begin the construction of the \begin{align*}45^\circ45^\circ90^\circ \end{align*} triangle, construct line segment \begin{align*}AB\end{align*} and a perpendicular to \begin{align*}AB\end{align*} at \begin{align*}A\end{align*}.
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*}.
Can you explain why \begin{align*}AB = AC\end{align*} and why angle \begin{align*}ACB = \end{align*} angle \begin{align*}ABC\end{align*}?
Why are these two angles \begin{align*}45^\circ\end{align*} each?
Students should notice that the two angles must be equal, and angle \begin{align*}A\end{align*} is \begin{align*}90^\circ\end{align*}. Therefore, because the sum of the angles in a triangle is \begin{align*}180^\circ\end{align*}, the two angles must be \begin{align*}45^\circ\end{align*} each.
Measure the sides of the triangle. This verifies that \begin{align*}AB = AC\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?
Students should notice that \begin{align*}AC\end{align*} and \begin{align*}AB\end{align*} are equal, so the ratios will always remain the same.
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Date Created:
Feb 23, 2012Last Modified:
Nov 03, 2014If you would like to associate files with this section, please make a copy first.