The Pythagorean Theorem
The Pythagorean Theorem is a very important concept in mathematics. The theorem can be understood on different cognitive levels by students with varying experience. For instance, a middle school student may use the Pythagorean Theorem to find the sides of a right triangle, while an Geometry student in high school may use the distance formula derived from the Pythagorean Theorem to find the radius of a circle.
There have been may different proofs of the Pythagorean Theorem. I present a proof here that I believe can be explained, taught, and developed on all levels. I also believe that students can discover this proof of The Pythagorean Theorem from this essay and using Geometer' Sketchpad.
* Take any right triangle with sides of lengths a1, b1, and c1 as illustrated. Click here to manipulate the GSP illustration below.
Extend lines from points A and B, perpendicular to line c1 that are the same length as c1. Connect the end points of these perpendicular segments, c2 and c3, with a segment c4 as illustrated below.
Now, extend segment b1 to the right until you are "abeam" point D so you can connect this segment with a perpendicular line through point D as illustrated below. The segment, a2, is the same length as segment a1 because the hypotenuses of the right triangles are the same length.
Now, connect the remaining points on the square in the same manner. Note that all of the "a" segments are equal in length, and all of the "b" segments are equal in length.
Click here to manipulate the GSP construction illustrated below.
Now, find the area of the pink square with side "c1". Also find the area of each of the blue triangles with legth "a" and height "b".
Notice in the illustration below that the area of the square is 2.21 square inches and the area of each triangle is 45 square inches.
We will find the areas algebraically, but you can also demonstrate this by using the GSP illustration shown below. Click here to go to the sketch using GSP.
Now, algebraically, find the total area of the drawing by adding the area of the pink square and all of the areas of the blue triangles.
The area of the pink square with side "c" is 
.
The area of each of the blue triangles is 
,
so the total area of all 4 blue triangles is 
.
So, the total area of the pink square and all of the blue triangles
is + 
.
Now, let's also find the area again using the fact that the sides of the total area... the "big" square have length "a + b".
So, to find the area, we can use the formula for finding the area of the square again to get
So, now we can set both areas equal to each other (because they are equal) and see what we get!
Total area by adding of the pink square and blue triangles:
Total area by squaring the side of the "big" triangle:
Now, subtract 2ab from both sides of the equation and you end up with:
which is the Pythagorean Theorem!
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