What is the greatest possible perimeter of a right angled triangle with integer lengths of one of the side has length 12?
A common right angled triangle for us is the triangle with sides (5,12,13).So the perimeter is 30.But I am not sure that it is the maximum or not.Please tell me the right answer with reason
Best Answer
The answer is $84$. It comes from the right triangle $(12,35,37)$.
You want the maximum of $12+a+b$ where $a$ and $b$ are such that $12^2+a^2=b^2$. This is equivalent to $(b-a)(b+a)=144$. Making the list of all possible ways of expressing $144$ as the product of two numbers both of which are even, you will find that the list is rather short. My answer comes from the fact that $144=2\times72$. For this decomposition, you get $a=35$ and $b=37$.