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If a right-angled triangle has integer sides then which of the following is necessarily an integer?

(A) Area

(B) Circumradius 

(C) In-radius 

(D) None of these 

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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.

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