Pythagoras and its converse

The hypotenuse is opposite the right angle

In a right triangle, if the perpendicular sides have lengths a,b and the hypotenuse has length c, then a² + b² = c². The three squares are areas built on the sides; the hypotenuse’s square has the combined area of the other two.

For perpendicular sides 6 cm and 8 cm, c² = 36 + 64 = 100, so c = 10 cm. Use the positive root for a length. Adding 6 and 8 would follow two sides, not the straight hypotenuse.

A missing shorter side needs subtraction

If the hypotenuse is 13 cm and one shorter side is 5 cm, the other satisfies b² = 13²-5² = 144, so b = 12 cm. Identify the hypotenuse before choosing addition or subtraction. The result must be shorter than 13 cm.

The converse checks a right angle

For sides 7, 24 and 25, test the longest side: 7² + 24² = 625 = 25², so the triangle is right-angled opposite the 25-unit side. For 5, 6 and 8, 25 + 36 ≠ 64, so it is not right-angled. Valid triangle lengths alone do not guarantee a right angle.

Check your understanding 1 Choose the longest side first

A triangle has sides 9 cm, 12 cm and 15 cm. Decide whether it is right-angled. Then find the missing shorter side of a different right triangle with hypotenuse 10 cm and one shorter side 4 cm.

Your answer or drawing on paper

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9² + 12² = 225 = 15², so the first is right-angled. For the second, the side is square root of (10²-4²) = square root of 84 = 2 square root of 21 cm, about 9.17 cm. It is not square root of 116 because 10 is already the hypotenuse.

State the model’s right angle

A straight ladder 5 m long rests against a vertical wall, with its foot 1.4 m from the wall on level ground. The modelled height is square root of (25-1.96) = 4.8 m. This calculation assumes the wall and ground meet at 90°.