Question

The sides of a triangle are three consecutive integers and its inradius is four units. Determine
the circumradius.

Answer 1

@Mertsj Help me sir

Answer 2

inradius:
\[r = \frac{1}{2}\sqrt{\frac{(b + c - a)(a+c-b)(a+b-c)}{a + b + c}}\]
given r = 4, let a = n, b = n+1, c = n+2:
\[8=\sqrt{\frac{(n+3)(n+1)(n-1)}{3(n+1)}}\]
\[8=\sqrt{\frac{n^2+2n-3}{3}}\]
\[\sqrt{64}=\frac{\sqrt{n^2+2n-3}}{\sqrt{3}}\]
\[\sqrt{192}=\sqrt{n^2+2n-3}\]

Find n, then use:
\[r=\frac{abc}{4sR}\]
where s is the semiperimeter \(\frac{1}{2}(a+b+c)\) and R is the circumradius

Answer 3

Thank you Madam