Let A B C be a right-angled triangle with B=90^ . Let the length of the altitude B D be equal to 12 . What is the minimum possible length of A C , given that A C and the perimeter of triangle A B C are integers?

Step-by-step solution

a c=12 b Let a+c=I l integer a^2+c^2=R-24 b=b^2 aligned & D= 0 & I^2-48 b 0 & b^2+24 I 0 b 24 & Let b=k-12 k 36 & array l R=k^2-144 144=(k-I)(k+I) For k=37, I=35 I=25 is minimum value array aligned