If the perimeter of a right-angled triangle is 30 cm and the hypotenuse is 13 cm, then what is the area of the triangle?

The sum of the two legs of the triangle equals 30 cm minus 13 cm, which is 17 cm. Let the legs be a and b, so the Pythagorean theorem gives a squared plus b squared equals 13 squared, or 169. Squaring the sum of the legs gives 17 squared equals 289, which equals a squared plus b squared plus 2ab. Substituting 169 for a squared plus b squared gives 289 equals 169 plus 2ab, so 2ab equals 120. The area of the right-angled triangle is one half of ab, which is 30 cm squared.

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