October 2026 - Sides of a Right Triangle Study Guide

The theorem, and which letter is which

In a² + b² = c², a and b are the two legs — the sides that meet at the right angle — and c is the hypotenuse. The letters are not interchangeable: c must be the side opposite the right angle, whatever the diagram labels it.

That is the whole method for finding a missing side. Square the two you know, then either add (if you want the hypotenuse) or subtract (if you want a leg), and take the square root.

The two directions

Deciding which case you are in before calculating prevents the only real error in this topic.

1. Find the right angle
2. The side opposite it is c
3. Missing c? Add the squares
4. Missing a leg? Subtract from c²
5. Square root

Finding a leg, not the hypotenuse

Triples worth recognising

These whole-number sets appear constantly, and spotting one saves the arithmetic entirely.

• 3, 4, 5 — and its multiples: 6, 8, 10; 9, 12, 15; 30, 40, 50.
• 5, 12, 13 — and 10, 24, 26.
• 8, 15, 17.
• 7, 24, 25.

Testing whether a triangle is right-angled

The theorem works in reverse. Given three sides, square them all: if the two smaller squares sum to the largest, the triangle has a right angle. For 6, 8 and 11, 36 + 64 = 100, which is not 121, so it does not.

Word problems use this for diagonals — the shortest distance across a rectangular room, the bracing on a rectangular frame, the straight-line distance when someone walks three blocks north and four blocks east.

Terms to know

Right triangle

A triangle containing one 90-degree angle.

Hypotenuse

The side opposite the right angle. Always the longest.

Leg

Either of the two sides that meet at the right angle.

Pythagorean theorem

a² + b² = c², relating the legs to the hypotenuse.

Pythagorean triple

Three whole numbers that satisfy the theorem, such as 3, 4, 5.

Square root

The number that, multiplied by itself, gives the value under the root.

What the TEAS asks most

The same core ideas appear in different wording. If you can answer these without stopping to think, you have what this section requires.

Key points

• a² + b² = c², where c is the hypotenuse — opposite the right angle, always longest.
• Add the squares to find the hypotenuse; subtract to find a leg.
• Check plausibility: the hypotenuse must exceed both legs.
• Recognise 3-4-5, 5-12-13, 8-15-17 and 7-24-25 and their multiples.
• Diagonal-across-a-rectangle problems are Pythagoras in disguise.