Triangles and the Pythagorean theorem

What every triangle obeys

Three facts hold for every triangle, whatever its shape.

1. The angles add to $180^\circ$.
2. Bigger sides face bigger angles. The longest side is opposite the largest angle, and the shortest side opposite the smallest. Equal sides face equal angles: in an isosceles triangle the two angles at the base are equal, and in an equilateral triangle every angle is $60^\circ$.
3. The triangle inequality. Each side is shorter than the other two together. If two sides are $a$ and $b$ with $a \ge b$, the third side $x$ satisfies

$$a - b < x < a + b$$

Why strict? If $x$ were exactly $a + b$, the two shorter sides laid end to end would only just reach, and the "triangle" would flatten into a straight segment with no area. Questions that ask how many whole-number lengths are possible depend on getting those two endpoints excluded.

Area: base times height, with the right height

$$\text{area} = \frac{1}{2} \times \text{base} \times \text{height}$$

Any side can be the base. The height is the perpendicular distance from the opposite vertex to the line containing that base. In a right triangle the two legs are a base and its height. In an obtuse triangle the height to one of the short sides falls outside the triangle, onto that side extended; the formula still works.

Because one triangle has one area, the three base–height pairs all give the same product. That is a useful tool: once you know the area, you can find the height to any other side by dividing twice the area by that side.

A slanted side is a height only if it is perpendicular to the base. Multiplying two sides that meet at an angle other than $90^\circ$ gives too large an area.

The Pythagorean theorem

In a right triangle with legs $a$ and $b$ and hypotenuse $c$ (the side opposite the right angle, always the longest side):

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

The theorem works in both directions. If three sides satisfy $a^2 + b^2 = c^2$, the angle opposite $c$ is a right angle. Comparing $c^2$ with $a^2 + b^2$ (where $c$ is the longest side) also sorts every triangle:

Longest side $c$	The angle opposite $c$ is	The triangle is
$c^2 < a^2 + b^2$	less than $90^\circ$	acute
$c^2 = a^2 + b^2$	exactly $90^\circ$	right
$c^2 > a^2 + b^2$	more than $90^\circ$	obtuse

The intuition: hold sides $a$ and $b$ fixed and open the angle between them. The wider the angle, the longer the third side; at exactly $90^\circ$ it equals $\sqrt{a^2 + b^2}$.

Whole-number right triangles worth knowing

Some side triples come up because they keep the arithmetic clean. Knowing them saves a square root.

Triple	Check
$3, 4, 5$	$9 + 16 = 25$
$5, 12, 13$	$25 + 144 = 169$
$8, 15, 17$	$64 + 225 = 289$
$7, 24, 25$	$49 + 576 = 625$

Any multiple of a triple is a triple too: $6, 8, 10$ and $15, 20, 25$ are both $3, 4, 5$ scaled up. Before you square anything, check whether the given sides are a multiple of a known triple. A leg of $10$ with a hypotenuse of $26$ is $5, 12, 13$ doubled, so the other leg is $24$.

Make sure the triple is in the right order. Legs of $5$ and $13$ do not give a hypotenuse of $12$: the hypotenuse is the longest side, so it would be $\sqrt{25 + 169} = \sqrt{194}$.

Where right triangles hide

Many questions do not show a right triangle; you build one. A rectangle's diagonal splits it into two right triangles. The height of an isosceles triangle drawn to its base cuts the base in half and makes two right triangles. A distance between two points on a grid is the hypotenuse of a triangle whose legs are the horizontal and vertical gaps. When a length is missing, look for a right angle you can create.