Question

The sides of a right triangle are in arithmetic progresssion. If the ratio of the perimeter and the area of that triangle is 2:1, then find the sides of the triangle ?​

Answer 1

I suspect that this question was supposed to specify a right triangle, because (as Gopal Menon noted) it is not true in general.

For a right triangle, suppose we have side lengths  a,a+h,a+2h , with  h>0 . We can rescale the triangle so that  a=1  (and the “new” value of h is  h/a ); then by the Pythagorean Theorem,

12+(1+h)22+2h+h200=(1+2h)2=1+4h+4h2=3h2+2h−1=(3h−1)(h+1)

The only positive solution is  h=13 , so the side lengths of the scaled triangle are  1,43,53 , which puts them in the ratio 3:4:5.