a special right triangle
Two right triangles come up so often that their side-ratios are worth memorising once and for all: the 45-45-90 triangle and the 30-60-90 triangle. Because their angles are fixed, all triangles of each kind are similar, so the sides always come in the same fixed proportion no matter the size. Knowing these lets you write down a missing side instantly, with no calculator and no trigonometry.
The 45-45-90 triangle is half of a square cut along its diagonal: two equal legs and a right angle, so the two acute angles are each 45 degrees. Its sides are in the ratio 1 : 1 : sqrt(2), meaning the hypotenuse is sqrt(2) times a leg. The 30-60-90 triangle is half of an equilateral triangle cut down the middle. Its sides are in the ratio 1 : sqrt(3) : 2, where 1 is the side opposite the 30-degree angle (the short leg, half the hypotenuse), sqrt(3) is opposite the 60-degree angle, and 2 is the hypotenuse. To use them, match the side you know to its place in the ratio and scale.
These ratios come straight out of the Pythagorean theorem (for the 45-45-90, 1^2 + 1^2 = 2 so the hypotenuse is sqrt(2)) and are the integer-and-surd backbone of trigonometry — they are exactly where sin 30 = 1/2, cos 45 = sqrt(2)/2, and tan 60 = sqrt(3) come from. An honest reminder: the numbers 1, sqrt(2), sqrt(3), 2 are ratios, not the actual lengths; in a real problem you multiply them all by whatever the real scale turns out to be.
A square has side 5; its diagonal is a 45-45-90 hypotenuse, so diagonal = 5·sqrt(2). In a 30-60-90 triangle with hypotenuse 10, the short leg (opposite 30) is 5 and the long leg (opposite 60) is 5·sqrt(3).
Fixed ratios 1 : 1 : sqrt(2) and 1 : sqrt(3) : 2 give missing sides at a glance.
In the 30-60-90 triangle, sqrt(3) is opposite the 60-degree angle, not the 30. Mixing up which leg is the short one (opposite 30, the half-hypotenuse) and which is sqrt(3) times longer is the usual mistake — the side opposite the bigger angle is always the longer one.