Right triangles and the Pythagorean theorem

In a right triangle the squares of the two sides about the right angle add up to the square of the hypotenuse¹. We confirm it for the triangle with vertices $A(-3, 0)$ , $B(3, 0)$ and $C(0, 3)$ .

Confirming the right angle

We have $\overrightarrow{CA} = (-3, -3)$ and $\overrightarrow{CB} = (3, -3)$ , whose dot product is $(-3)(3) + (-3)(-3) = 0$ . The two sides are perpendicular.

Confirming the theorem

The lengths of the three sides are $CA = CB = \sqrt{3^2 + 3^2} = 3\sqrt{2}$ and $AB = 6$ .

CA^2 + CB^2 = 18 + 18 = 36 = 6^2 = AB^2

The two sides about the right angle are equal, so this is a right isosceles triangle with sides in the ratio $1 : 1 : \sqrt{2}$ . The number $\sqrt{2}$ is the one that appears as the diagonal of a square, and this triangle is a square cut in half along that diagonal.

Combinations that come out whole

There are combinations in which all three sides are integers, called Pythagorean triples².

$a$	$b$	$c$	$a^2 + b^2 = c^2$
$3$	$4$	$5$	$9 + 16 = 25$
$5$	$12$	$13$	$25 + 144 = 169$
$8$	$15$	$17$	$64 + 225 = 289$

The converse also holds

In a triangle whose sides satisfy $a^2 + b^2 = c^2$ , the angle opposite $c$ is a right angle. That is the converse of the Pythagorean theorem.

A proof by area

Inside a square of side $a + b$ , arrange four right triangles with legs $a$ and $b$ like the sails of a windmill; a square of side $c$ is left in the middle. Counting the total area in two ways gives the following.

\begin{align*} (a + b)^2 &= 4 \times \frac{ab}{2} + c^2 \\ a^2 + 2ab + b^2 &= 2ab + c^2 \\ a^2 + b^2 &= c^2 \end{align*}

Deciding the type of an angle

The lengths of the three sides also reveal the type of the angle.

Condition	The angle opposite the longest side
$a^2 + b^2 > c^2$	acute
$a^2 + b^2 = c^2$	right
$a^2 + b^2 < c^2$	obtuse

For $5$ , $6$ and $8$ , say, $25 + 36 = 61$ falls short of $8^2 = 64$ , so the angle opposite the longest side is obtuse.

Relation to the distance formula

The formula for the distance between two points in the coordinate plane is this theorem itself.

\sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

It builds a triangle whose legs are the difference in $x$ and the difference in $y$ , and asks for the hypotenuse. That is why the theorem shows its face everywhere in coordinate calculations.

The three lines on the graph are the three sides, and the large dots are the three vertices.

Pythagorean theorem, Wikipedia
Pythagorean triple, Wikipedia