The sides of a triangle are three consecutive integers and its inradius is 4. Find the circumradius.
For a triangle with sides a, b, c:
Inradius formula: r = (1/2)*sqrt[(a+b-c)*(a-b+c)*(-a+b+c)/(a+b+c)]
Circumradius formula: R = (a*b*c)/(2*r*(a+b+c))
So let the sides of the triangle be x-1, x, and x+1. Then these formulas reduce down to:
r = (1/2)*sqrt[(x-2)*(x+2)/3]
R = (x^2-1)/(6*r)
We are given r=4. Then 4=(1/2)*sqrt[(x-2)*(x+2)/3] after squaring each side reduces to 196=x^2.
Then plugging these values to find R yields R = (196-1)/(6*4) = 8.125.
Solution
