Find a shorter side

Objective. Rearrange Pythagoras to find a missing leg.

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Learn: the key idea

When the hypotenuse and one shorter side of a right-angled triangle are known, Pythagoras' theorem is rearranged to find the missing shorter side using a² = c² − b². Square the hypotenuse and the known side, subtract the known side's square from the hypotenuse's square, then take the square root of the result. Example: with hypotenuse 13 cm and one side 5 cm, compute 13² − 5² = 169 − 25 = 144, then √144 = 12 cm. It helps to check that this new side, combined with the original, still satisfies a² + b² = c².

Worked example

A right-angled triangle has a hypotenuse of 17 cm and one shorter side of 8 cm. Find the length of the other shorter side.

1 $Square the hypotenuse and the known side: 17² = 289 and 8² = 64.$
2 $Subtract the known side's square from the hypotenuse's square: 289 - 64.$
3 $This gives 289 - 64 = 225.$
4 $Take the square root: \sqrt225 = 15 cm.$

Answer: 15 cm

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Common mistake. Pupils often add the squares instead of subtracting when finding a shorter side, treating it the same as finding the hypotenuse; remember that a missing shorter side needs the hypotenuse's square minus the known side's square.

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