The Hypotenuse of a Right-Angled Triangle

Two tests that always agree

The hypotenuse is the longest side, and it is the side opposite the right angle. These are not two separate facts to remember; they always identify the same side, because in any triangle the largest angle faces the largest side, and the right angle is necessarily the largest when one is present.

In practice, find the right angle first and follow it across the triangle. That works on a diagram even when no lengths are marked, whereas the longest-side test needs numbers.

It is not simply the sloping side

Most textbook triangles are drawn with the right angle at the bottom left and the hypotenuse sloping up to the right, and students quietly learn the picture rather than the rule. Rotate the same triangle a quarter turn and the hypotenuse becomes vertical.

The mistake bites hardest inside larger diagrams. A right-angled triangle cut from a rectangle, or one of several triangles inside a trapezium, will often have its hypotenuse drawn horizontally while some other line slopes across the page. Locate the right angle in that particular triangle every time.

A triangle with no right angle has no hypotenuse at all. The longest side of a scalene or obtuse triangle simply is not called one, and Pythagoras' theorem does not apply to it.

Why the label decides the method

With Pythagoras, you add the squares to find the hypotenuse and subtract to find a shorter side. Labelling the sides before calculating is what tells you which of those to do, and it is the single most common place marks are lost in the topic.

Trigonometry depends on it just as heavily. Sine is opposite over hypotenuse and cosine is adjacent over hypotenuse, so labelling the wrong side swaps the two ratios. There is a built-in check: for an acute angle, sine and cosine are always less than 1, so an answer above 1 means the hypotenuse was put in the wrong place.