If the lengths of the sides of a right angled triangle are in A.P., then show that their ratio is 3 : 4 : 5.

Solution

Let a - d, a, a + d, with d>0 be the lengths of sides of a right angled triangle.

∴ a + d, the largest side must be hypotenuse.
So, by pythagorus theorem
$left parenthesis straight a plus straight d right parenthesis squared space equals space straight a squared plus left parenthesis straight a minus straight d right parenthesis squared$ or $straight a squared plus straight d squared plus 2 ad space equals space space straight a squared plus straight a squared plus straight d squared minus 2 ad$
or $space space straight a squared space equals space 4 ad$ or a = 4d