Proof of the Pythagorean Theorem (Using Similar Triangles)

The famous Pythagorean Theorem says that, for a right triangle

- (Length of Leg_A)² + (Length of Leg_B)² = (Length of Hypotenuse)²

Or more commonly, a² + b² = c²

There are many, many proofs of the Pythagorean Theorem. Many of them involve a picture where triangles and squares get moved around. Here's one that uses similar triangles.

Demonstration

Prove the Pythagorean Theorem using similar triangles.

First let's take our triangle and draw a line

perpendicular to side AB
that goes through point C.

(lines that do this are called altitudes)

We'll label the point where our altitude intersects line AB as point D

Now we have a lot of sides to label

Our original triangle had 3 sides

The side across from ∠B will have length b
The side across from ∠A will have length a
The side across from ∠C will have length c

Point D splits AB into two segments, AD and DB

- AB will have length d
- DB will have length e

Line CD also needs a length

- Line CD have length x

This now gives us three triangles

Now, what can we say about these 3 triangles?

Let's look at ACE and ABC. Both have a 90° angle, and ∠CAB and.∠CAE are the same angle. If they have two congruent angles, then by AA criteria for similarity, the triangles are similar. Making sure to write the similarity statement congruent angles corresponding, we can say.

ΔABC ~ ΔACE (Δ-shared angle -- other angle -- 90° angle)

Let's look at CEB and ABC. Both have a 90° angle, and ∠CBA and.∠CBE are the same angle. If they have two congruent angles, then by AA criteria for similarity, the triangles are similar. Making sure to write the similarity statement congruent angles corresponding, we can say.