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The Pythagorean Theorem is one of the great ideas in mathematics. The following proofs will take you through two of the over 300 different methods that have been discovered to show that given a right triangle ABC, with the right angle at C, the sum of the squares of the legs of the triangle equal the square of the hypotenuse.
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Step 1:
Construct a right triangle with the right angle at C.
Step 2:
Construct a square at each edge of the triangle. This can be done by any of several methods.
Step 3:
Extend lines along the sides of the squares, as show below:
Step 4:
Using A as a center point, construct a circle with radius AB. Similarly, construct a circle with B as a center point and radius AB. Find the points of intersection of these circles with the lines r and q (along the edges of the small squares). These intersection points are labeled D and F, as shown:
Note that a segment BF will be congruent (of same length) to segment AB, the hypotenuse of the right triangle. Similarly, a segment AD will be congruent (of same length) to segment AB.
Step 5:
The next step involves "shearing" the small squares. This will distort their shapes from squares into parallelograms, but their areas will remain the same. [Why? Good question.]
Looking only at the light blue square (with side a) and its "sheared" image in dark blue (CBFC'), the bottom edge is sheared into position FC', and a parallelogram CBFC' is formed. This parallelogram has the same area as the light blue square.
Repeat this process with the small light green square (with side b).
Step 6:
Shear the parallelograms along the line CC' until they meet line segment AB (and thus are quadrilaterals).
The resulting square, FDAB has the same area as the square of the hypotenuse. Why? Because its area must be BF times AB, and, as already noted, BF is congruent to AB.
Therefore, given a right triangle, the sum of the squares of its legs will always equal the square of its hypotenuse.
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Matthew C. Whitney
