Pythagorean Theorem

By

Wenda Peltz

The Pythagorean Theorem can explored in a number of different ways.  One such way is using GSP.  A simple triangle can be created using line segments in GSP.  There is no real need for the original triangle to be a right triangle.  Once the triangle is created, the vertices should be labeled.  An example of such a sketch using GSP can be seen below:

Once the triangle is created, the measure tools of GSP can be used to determine the lengths of the triangle and the measures of the angles.  The calculate tool can be used to create equations to square each length and to add the squares of each two sets of sides.  With all of the measurements and calculations showing on the screen, you can observe what happens to the calculations as the vertices are moved to change the measure of the angles.  An example of the needed calculations can be seen below.

As you move the vertices around you can study the measure of the angles and/or the relationship between the sum of the squares of two sides and the square of the remaining side.  Hopefully, you will notice that when any of the angle measures equal 90 that the sum of the squares of two sides will equal the square of the remaining side.  An example is given below.

You should notice in the image above that the measure of angle BCA is equal to 90 degrees.  When this is true, the sum of the squares of sides AC and CB is 157.03, which is equal to the square of side AB.  This is a demonstration of the Pythagorean Theorem.

This topic could be explored further by manipulating the other vertices.  GSP could be used to draw actually squares that have sides equal to the lengths of the sides of the triangle and the areas of the squares could be compared as the triangle is manipulated.