Using Similarity to Prove the Pythagorean Theorem

The Pythagorean Theorem is one of the most interesting theorems for two reasons: First, it’s very elementary; even high school students know it by heart. Second, it has hundreds of proofs. The proof below uses triangle similarity.

Pythagorean Theorem

In a right triangle with side lengths a and b and hypotenuse c,  the following equation always holds:

c^2 = a^2 + b^2.

Proof

In the figure above, triangle BDA is similar to triangle ADE.  In similar triangles, the corresponding sides are proportional. It follows that

\displaystyle\frac{BD}{AD} = \frac{AD}{DE}

which is equivalent to

\displaystyle\frac{c + a}{b} = \frac{b}{c-a}.

Cross multiplying, we have c^2 - a^2 = b^2. Adding a^2 to both sides of the equation, we  have c^2 = a^2 + b^2, which is what we want to prove.

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2 Comments

  1. Nice proof! I’ve seen a version of this one where the hypotenuse of the right triangle is the diameter of the circle. I don’t think I’ve seen the idea of using the hypotenuse as the radius like this.

    Reply
    • It’s actually a derivative of one of the proofs I’ve seen. I’m searching for the same proof on the net. It could be an addition to the hundreds of proofs.

      Reply

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