Understanding and applying the theorem
Right-angled triangles and the hypotenuse
What is a right-angled triangle?
A right-angled triangle is a triangle that has one right angle, i.e. an angle of 90 degrees. This right angle is often marked with a small square on diagrams.
Key terminology
In a right-angled triangle, the three sides do not all have the same name:
| Side | Definition |
|---|---|
| Hypotenuse | The longest side, always opposite the right angle |
| Sides of the right angle | The other two sides, which form the right angle between them |
The hypotenuse is easy to identify: simply locate the right angle on the diagram, and the side opposite it is the hypotenuse.
Example
Consider a right-angled triangle ABC with the right angle at A (the right angle is at vertex A). Then:
- Side [BC] is the hypotenuse (it is opposite vertex A)
- Sides [AB] and [AC] are the sides of the right angle
Why this terminology is essential
The rest of the lesson relies entirely on this identification. Before using a formula, you must always start by answering the question: where is the right-angled corner, and which side is therefore opposite it?
Common pitfall
Never confuse the hypotenuse with a side chosen at random. Many pupils choose the side that ‘seems’ longest in a poorly proportioned diagram. The only reliable rule is: the hypotenuse is ALWAYS the side opposite the right-angled corner, never one of the two sides forming that angle.