The Pythagorean Theorem and Its Converse

By the end of this lesson, you will be able to prove the Pythagorean Theorem using similar triangles, apply it and its converse to find missing side lengths and classify triangles, and use Pythagorean triples fluently.

The Big Idea

The Pythagorean Theorem is one of the most proven theorems in mathematics — over 370 different proofs exist. In a geometry course, the most illuminating proof uses similar triangles, revealing that the theorem is really a statement about proportions in right triangles. The converse is equally powerful: it converts a measurement question into a classification question.

The Theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$:

$$a^2 + b^2 = c^2$$

The hypotenuse is always opposite the right angle — the longest side.

Proof Using Similar Triangles

Draw altitude $\overline{CD}$ from the right angle $C$ to the hypotenuse $\overline{AB}$ in right triangle $\triangle ABC$. This creates two smaller triangles: $\triangle ACD$ and $\triangle CBD$.

$\triangle ABC \sim \triangle ACD \sim \triangle CBD$ (all by AA — they each share an angle with $\triangle ABC$ and have a right angle).

From $\triangle ABC \sim \triangle ACD$:

$$\frac{AC}{AB} = \frac{AD}{AC} \implies AC^2 = AB \cdot AD$$

From $\triangle ABC \sim \triangle CBD$:

$$\frac{BC}{AB} = \frac{DB}{BC} \implies BC^2 = AB \cdot DB$$

Adding:

$$AC^2 + BC^2 = AB(AD + DB) = AB \cdot AB = AB^2$$

So $a^2 + b^2 = c^2$. ∎

The Converse

If $a^2 + b^2 = c^2$, then the triangle with sides $a$, $b$, $c$ is a right triangle (with the right angle opposite $c$).

Extension:

• $a^2 + b^2 > c^2$ → acute triangle
• $a^2 + b^2 < c^2$ → obtuse triangle

(where $c$ is the longest side)

Pythagorean Triples

Sets of positive integers satisfying $a^2 + b^2 = c^2$:

Triple	Verification
$3, 4, 5$	$9 + 16 = 25$ ✓
$5, 12, 13$	$25 + 144 = 169$ ✓
$8, 15, 17$	$64 + 225 = 289$ ✓