If the hypotenuse of a right angled triangle is $41\ cm$ and the area of the triangle is $180\ sq.cm$, find the difference between the lengths of the legs of the triangle

Let the legs be a and b. The area gives the equation one half of a times b equals 180, so a times b equals 360. Pythagoras theorem gives a squared plus b squared equals 41 squared, which is 1681. The difference between the legs squared is (a squared plus b squared) minus 2ab, calculated as 1681 minus 720, which equals 961. Taking the square root of 961 gives the difference between the legs as 31 cm.

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