Answer :

A = √ s(s-a)(s-b)(s-c)


If the area is an integer


Then [s(s-a)(s-b)(s-c)] should be proper square


If s = Then s = = 24


Hence ;


A = √ 24(24-a)(24-b)(24-c)


If side of triangle are


a = 21 and b + c = 27


let c be smallest side


then b = 27-c


√ 24(24-21)(24-27 + c)(24-c)


√ 24 × 3 × (c-3)(24-c)


√ 2 × 2 × 2 × 3 × 3 × (c-3)(24-c)


2 × 3√2(c-3)(24-c)


6√2(c-3)(24-c)


the value of [2(c-3)(24-c)] must be a perfect square for area to be a integer


For getting square 2(c-3) should be equal to (24-c)


2(c-3) = (24-c)


2c-6 = 24-c


2c + c = 24 + 6


3c = 10


c = 10; b = 27-c = 27-10 = 17


Hence the size of smallest size is 10.


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