Pythagorean Theorem

One of the oldest and most famous mathematical concepts is the Pythagorean theorem. Named after the ancient Greek mathematician Pythagoras, this theorem establishes a fundamental relationship between the sides of a right-angled triangle.

Right-Angled Triangle

Definition Right-Angled Triangle

A right-angled triangle is a triangle that has one right angle (90°).

The two sides that form the right angle are called the legs of the triangle.
The side opposite the right angle, which is also the longest side, is called the hypotenuse.

Pythagorean Theorem

Theorem Pythagorean Theorem

For any right-angled triangle with legs of length $\textcolor{colordef}{a}$ and $\textcolor{colorprop}{b}$ and a hypotenuse of length $\textcolor{olive}{c}$, the square of the hypotenuse is equal to the sum of the squares of the legs:$$ \textcolor{colordef}{a^2} + \textcolor{colorprop}{b^2} = \textcolor{olive}{c^2}. $$This means the area of the square on the hypotenuse is equal to the sum of the areas of the squares on the other two sides.

Proof

The geometric proof involves rearranging four identical right-angled triangles within a square of side length $a + b$.
1. In the first arrangement, the four right-angled triangles form a smaller square in the center of the larger square:
2. In the second arrangement, the four right-angled triangles form two smaller squares within the larger square:
3. Since the area not covered by the triangles is equal in both arrangements, we have:$$a^2 + b^2 = c^2.$$
Algebraic proof: Using four copies of the same triangle arranged symmetrically around a central square with side $c$, inside a larger square with side $a + b$:The larger square has side $a + b$, with area $(a + b)^2$. The four triangles and the central square of side $c$ have the same total area as the larger square:$$\begin{aligned}(a + b)^2 &= \overbrace{c^2}^{\text{central square area}} + 4 \times \overbrace{\frac{ab}{2}}^{\text{triangle area}} \\ a^2 + 2ab + b^2 &= c^2 + 2ab \\ a^2 + b^2 &= c^2.\end{aligned}$$

$$\begin{aligned}\textcolor{colordef}{4^2} + \textcolor{colorprop}{3^2} &= \textcolor{olive}{5^2} \\ \textcolor{colordef}{16} + \textcolor{colorprop}{9} &= \textcolor{olive}{25}\end{aligned}$$This diagram provides a visual representation of the Pythagorean theorem.

Example

Find the length of the hypotenuse, $x$.

Answer

By the Pythagorean theorem:$$ \begin{aligned} a^2 + b^2 &= c^2 \\ 3^2 + 5^2 &= x^2 \\ 9 + 25 &= x^2 \\ 34 &= x^2 \\ x &= \sqrt{34} \, \text{cm}\end{aligned} $$The hypotenuse has a length of $\sqrt{34}$ cm (approximately 5.83 cm).

Example

Find the length of the unknown leg, $y$.

Answer

By the Pythagorean theorem, with $c = 13$ being the hypotenuse:$$ \begin{aligned} a^2 + b^2 &= c^2 \\ y^2 + 5^2 &= 13^2 \\ y^2 + 25 &= 169 \\ y^2 &= 169 - 25 \\ y^2 &= 144 \\ y &= \sqrt{144} = 12 \, \text{m}\end{aligned} $$The unknown leg has a length of 12 m.

Verifying Right-Angled Triangles

Theorem Converse of the Pythagorean Theorem

If a triangle has side lengths $a$, $b$, and $c$, where $c$ is the longest side, and they satisfy $a^2 + b^2 = c^2$, then the triangle is a right-angled triangle, and the right angle is opposite the longest side, $c$.

Example

Is a triangle with side lengths 5 cm, 12 cm, and 13 cm a right-angled triangle?

Answer

We check if the sum of the squares of the two shorter sides equals the square of the longest side. The longest side is 13 cm.

$ 5^2 + 12^2 = 25 + 144 = 169 $
$ 13^2 = 169 $

Since $5^2 + 12^2 = 13^2$, the triangle is right-angled by the converse of the Pythagorean theorem.

Theorem Contrapositive of the Pythagorean Theorem

If a triangle has side lengths $a$, $b$, and $c$, where $c$ is the longest side, and the sum of the squares of the two shorter sides is not equal to the square of the longest side ($a^2 + b^2 \neq c^2$), then the triangle is not a right-angled triangle.

Proof

This theorem is the contrapositive of the original Pythagorean theorem. Since a statement and its contrapositive are logically equivalent, the contrapositive is also true.

Example

Is a triangle with sides of lengths 5, 8, and 9 right-angled?

Answer

The two shorter sides are 5 and 8, and the longest side is 9.

$ 5^2 + 8^2 = 25 + 64 = 89 $
$ 9^2 = 81 $

Since $89 \neq 81$, the triangle is not right-angled by the contrapositive of the Pythagorean theorem.