**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 a**^{2} + b^{2} = c^{2}, we have following cases:

**(i)** If a and b are both odd or both even, then a^{2} + b^{2} is even ⇒ c^{2} 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 a^{2} + b^{2} is odd ⇒ c^{2} 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.