The Pythagorean Theorem and its many proofs

Pythagorean Theorem
In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the legs.

a² + b² = c²

There are several methods to prove the Pythagorean Theorem. Here are a few:

Method One:

Given triangle ABC, prove that a² + b² = c².

First construct a perpendicular from C to segment AB.

In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments (e and f).  The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg, therefore

and   .

If I cross multiply each ratio I get:

and

then adding a and b yields:

,

factoring out c on the right side gives:

,

and looking at the above diagram:

so by substitution we get :

Method Two:

For this proof I will use the below diagram:

Notice that the area of the large square could be expressed in two ways:

or

Summing the area of the four triangles () and the area of the smaller square ().

Since these both represent the area of the larger square we will set them equal to one another.

=

Use algebra to simplify the above equation.

=

If we subtract 2ab from both sides of the equation we obtain:

The Pythagorean Theorem.

Method Three:

For this proof I will use the below diagram:

The area of the above trapezoid can also be expressed in two ways.  One is by using the formula for the area of a trapezoid:

Let’s use the diagram above to fill in the formula for the area of a trapezoid:

The second way to express the area of the above trapezoid is by summing each geometric figure.

Since these two expressions are the area of the above trapezoid we will set them equal to each other and use algebra to simplify the equation.

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