Correct option (C) In-radius
See Fig.
Let triangle be right-angled at C. Then area = 1/2ab, circumradius R = c/2 which are not necessarily integers. Again in square ONCL, NC = OL = r. We have
c = AB = AM + BM = AL + BN
= b - r + a - r ⇒ r = a + b - c/2
As a2 + b2 = c2, we have following cases:
(i) If a and b are both odd or both even, then a2 + b2 is even ⇒ c2 is even Therefore, c is even and so (a + b) – c is even.
(ii) If one of a and b is odd and the other even, then a2 + b2 is odd ⇒ c2 is odd
Therefore, c is odd and so (a + b) – c is even.
So, in every case if a, b, c are integers, we have r = c + a - b/2 = integer.
