Pythagorean Theorem: Proof

How to prove the Pythagorean theorem: formula and its proof.

Proof

Formula

For a right triangle,
a2 + b2 = c2.

This is the Pythagorean theorem.

Let's see the proof of the Pythagorean theorem.

Proof

Here's a right triangle.
The legs are a and b.
And the hypotenuse is c.

The plane angle, the dot angle, and the right angle
are the interior angles of the triangle.

So m∠[plane] + m∠[dot] + 90 = 180.

Draw 4 triangles like this.

Recall that
m∠[plane] + m∠[dot] + 90 = 180.

See each side of the whole figure.
The plane angle, the dot angle, and the brown angle
form a line.

So m∠[plane] + m∠[dot] + m∠[brown] = 180.

So m∠[brown] = 90.

So the brown angles are the right angles.

The whole figure is a square.

Its side is (a + b).

So the area of the whole square is
(a + b)2.

Square: Area

This is equal to ...

See one of the right triangle.

The base is b.
The height is a.
So the area of the triangle is
(1/2)ba = (1/2)ab.

There are 4 triangles.

So the sum of the areas of these 4 triangles is
4⋅(1/2)ab.

Plus ...

The center figure is a square.

Its side is c.

So the area of the center square is
c2.

So
(a + b)2 = 4⋅(1/2)ab + c2.

(a + b)2 = a2 + 2ab + b2

Square of a Sum: (a + b)2

4⋅(1/2)ab = 2ab

Cancel +2ab on both sides.

Then a2 + b2 = c2.

So a2 + b2 = c2.

This is the proof of the Pythagorean theorem.