Permutations and Combinations

Permutations and Combinations

Question

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

Answer

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

∴  Sides are 4d - d, 4d, 4d + d              or          3d, 4d, 5d
Hence, the sides are in the ratio 3:4:5.

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