This activity is designed to illustrate the way the area of isosceles triangles with set side lengths changes as the base of the triangle is changed. Students will consider such concepts such as length, angle measurements, and area formulas in their exploration of these ideas.

Task 1: Determining Base Length and Angle Measures For Greatest Area
1) With the open GeoGebra applet, deselect all of the check boxes on the left side of the screen.
2) Click on the slider labeled “Base.”
3) Use the arrow keys to extend or shorten the base of the isosceles triangle.
4) Make a prediction:
At what base length does the area of the isosceles triangle appear to be the greatest?
Approximate the angle measures of Angle A and Angle B at this base length.
5) Click on the check box with description “Show trace of area ABC vs. base length.”
6) Click on the slider labeled “Base.”
7) Use the arrow keys to extend or shorten the base of the isosceles triangle.
8) Record the base length that maximizes the area of the isosceles triangle with two congruent side lengths of 5 cm.
9) Click on the check box with description “Display Angle A and Angle B.”
10) Record the angle measures that maximize the area of the isosceles triangle with two congruent side lengths of 5 cm.
Task 2: Analyzing the Dynamic Properties of the Area vs. Base Length Graph
1) With the open GeoGebra applet, deselect all of the check boxes on the left side of the screen.
2) Click on the slider labeled “Base.”
3) Using the arrow keys, move the base length to 0.
4) Click on the check box with description “Show trace of area ABC vs. base length.”
5) Using the arrow keys, change the base length, taking note of the trace of the area vs. base length.
a) What did you notice when the base length was 0?
b) Describe the slope of the graph for base lengths between (0, 7)
c) Describe the slope of the graph for base lengths between (7, 10)
d) Why is the area of the triangle listed as undefined for base lengths of 10 or greater?
Task 3: Creating an Equation to Model the Relationship Between Area and Base Length
1) What is the formula for the area of a triangle?
2) How can we use the Pythagorean theorem to solve for the triangle’s height?
3) Solve for the height of the isosceles triangle as a function of base length.
4) Use this value for height to determine the area of the isosceles triangle as a function of the base length.
5) Check your solution by evaluating your equation at the base length and area you recorded in Task 1.
Extension: For Calculus classes, have students find the derivative of their area function to find the maximum values.