In a right-angled triangle the longest side is opposite the right angle. It is called the hypotenuse. Pythagoras' theorem says the squares of the two shorter sides add up to the square of the hypotenuse.

a² + b² = c²

1. shorter sides 6 cm and 8 cm, hypotenuse x
2. x² = 6² + 8² = 36 + 64 = 100
3. x = √100 = 10 cm

When you know the hypotenuse, subtract: the shorter side squared is the hypotenuse squared take away the other side squared.

1. hypotenuse 13 cm, one side 5 cm
2. x² = 13² − 5² = 169 − 25 = 144
3. x = √144 = 12 cm

Always check: the hypotenuse must be the longest side. If your "shorter side" comes out longer than the hypotenuse, you added when you should have subtracted.

If a² + b² = c² for the three sides, the triangle has a right angle. Builders use this to check corners: 30 cm, 40 cm and 50 cm make a right angle because 900 + 1600 = 2500.

For two points, draw the right-angled triangle under the line between them. The sides are the difference in x and the difference in y. From (−2, 1) to (4, 9): across 6, up 8, so the distance is √(36 + 64) = 10.

A common slip is forgetting the square root at the end: c² = 100 means c = 10, not 100.