Special Right Triangles

Two right triangles come up so often that their side ratios are worth memorising.

45-45-90 (half a square): legs are equal, hypotenuse is a leg times √2.
So legs x, x and hypotenuse x√2.

30-60-90 (half an equilateral triangle): sides are x, x√3, 2x.

• shortest side (opposite 30°): x
• middle side (opposite 60°): x√3
• hypotenuse (opposite 90°): 2x

The rule for keeping them straight: always start from the *shortest* side in a 30-60-90. Double it for the hypotenuse, multiply by √3 for the middle.

Working backwards means dividing, and that often needs rationalising. If the hypotenuse of a 45-45-90 is 10, each leg is 10/√2 = 5√2.

Both triangles are derived rather than memorised facts. The 45-45-90 is half a square cut along a diagonal: with legs x, the Pythagorean theorem gives hypotenuse x√2. The 30-60-90 is half an equilateral triangle of side 2x cut by an altitude, giving legs x and √(4x² − x²) = x√3.

The ratios are 1 : 1 : √2 and 1 : √3 : 2 respectively, with the 30-60-90 always ordered from the 30° side.

These supply the exact trigonometric values used throughout the rest of mathematics: sin 30° = ½, sin 45° = √2/2, sin 60° = √3/2, and the corresponding cosines and tangents. Knowing the triangles means never memorising that table separately.

Working from the hypotenuse back to a leg requires division, and answers should be rationalised: 10/√2 = 5√2. Leaving a radical in a denominator is the most common presentation error in the topic, and decimal approximations lose the exactness that makes these triangles worth using at all.