Question

If the sides of a right-angled triangle are in A.P. then prove that the ratio of the sides of that triangle is $3:4:5$.

Answer 1

Let the sides of the right-angled triangle are $a-d,a,a+d$. Where we let $a+d$ to be the hypotenuse of the right-angled triangle, with $a\ne 0,d>0$.

Then we've,

$(a-d{)}^2+a^2=(a+d{)}^2$

or, $2a^2-2ad+d^2=a^2+2ad+d^2$

or, $a^2=4ad$

or, $a=4d$. [ Since $a\ne 0$]

So the sides of the triangle are $3d,4d,5d$. So the ratio of the sides is $3:4:5$.