Proving the Pythagorean Theorem with Similarity

You have used $a^2 + b^2 = c^2$ since The Pythagorean Theorem, and seen why it works by rearranging areas. The similar triangles from Similar Right Triangles and the Geometric Mean give a short proof that uses no areas at all.

Why it works

Let right $\triangle ABC$ have legs $a = BC$ and $b = AC$ and hypotenuse $c = AB$. The altitude from $C$ cuts the hypotenuse into $p = AD$ and $q = DB$, with $p + q = c$. From $\triangle ACD \sim \triangle ABC$: $\frac{p}{b} = \frac{b}{c}$, so $b^2 = cp$. From $\triangle CBD \sim \triangle ABC$: $\frac{q}{a} = \frac{a}{c}$, so $a^2 = cq$. Add: $a^2 + b^2 = cq + cp = c(p + q) = c \cdot c = c^2$.

Worked example: The proof in two columns

Given: right $\triangle ABC$ with right angle at $C$ and altitude $\overline{CD}$. Prove: $a^2 + b^2 = c^2$.

1. $\triangle ACD \sim \triangle ABC$ and $\triangle CBD \sim \triangle ABC$ — AA (each shares an acute angle with $\triangle ABC$, and all have a right angle).
2. $\frac{p}{b} = \frac{b}{c}$ and $\frac{q}{a} = \frac{a}{c}$ — Corresponding sides of similar triangles are proportional.
3. $b^2 = cp$ and $a^2 = cq$ — Multiplication Property of Equality.
4. $a^2 + b^2 = cp + cq = c(p + q)$ — Addition Property; Distributive Property.
5. $p + q = c$ — Segment Addition Postulate.
6. $a^2 + b^2 = c^2$ — Substitution.

Answer: The Pythagorean theorem holds for every right triangle.

The area proof you saw before. Two very different arguments, the same theorem.

Key idea

In a right triangle, $a^2 = cq$ and $b^2 = cp$ (each leg squared is the hypotenuse times the adjacent piece). Adding them gives $a^2 + b^2 = c(p + q) = c^2$.

Worked example: The pieces add up

In a right triangle the altitude splits the hypotenuse $c = 10$ into $p = 6.4$ and $q = 3.6$. Find $a$ and $b$, and check $a^2 + b^2 = c^2$.

$b^2 = cp = 10 \cdot 6.4 = 64$, so $b = 8$.
$a^2 = cq = 10 \cdot 3.6 = 36$, so $a = 6$.
$36 + 64 = 100 = 10^2$.

Answer: $a = 6$, $b = 8$

The converse. If $a^2 + b^2 = c^2$ in a triangle, it is a right triangle. Build a right triangle with legs $a$ and $b$; its hypotenuse is $\sqrt{a^2 + b^2} = c$. The two triangles are congruent by SSS, so the original angle opposite $c$ is a right angle too.

Worked example: Using the converse

Is a triangle with sides $7$, $24$ and $25$ a right triangle?

$7^2 + 24^2 = 49 + 576 = 625 = 25^2$.

Answer: Yes, the angle opposite the side of length $25$ is a right angle.

Common mistake

Adding the wrong squares. The hypotenuse is always the longest side and sits alone on one side of the equation. For sides $5$, $12$, $13$: $5^2 + 12^2 = 13^2$, not $5^2 + 13^2 = 12^2$.

Proving the Pythagorean Theorem with Similarity

Practice

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