omair_bba wrote: Please tell where I am wrong with the concept..
(The length of any side of a triangle must be larger than the positive difference of the other two sides, but smaller than the sum of the other two sides.) For two given sides 12,13 the third side (e.g X) should be 1 < X > 25.

This rule applies to all triangles (right triangles and non-right triangles).
If we removed the "right triangle" restriction from the original question, then we could apply your rule. However, as it stands, the "right triangle" restriction limits our solution even further.

Cheers,
Brent

omair_bba wrote: Q: Two sides of right triangle are 12 & 13. Which of the following could be the length of the third side.
I. 5 II. 11 III. Sqrt 313

A) I only
B) II only
C) I & II only
D) I & III only
E) I,II & III

Since we're talking about a right triangle, we can apply the Pythagorean Theorem (a^2 + b^2 = c^2)

We must consider two cases:

case 1: the length of the hypotenuse (the longest side) is 13.
In this case, the missing side is a leg.
If we let x = the length of the missing side, we can write x^2 + 12^2 = 13^2
Solve to get x=5

case 2: the length of the hypotenuse (the longest side) is not 13.
Notice that 12 cannot be the length of the hypotenuse, since it would not be the longest side.
So, the missing side must be the hypotenuse.
If we let x = the length of the missing side, we can write 12^2 + 13^2 = x^2
Solve:
144 + 169 = x^2
313 = x^2
x = sqrt(313)

So, the missing side can have length 5 or sqrt(313)

The answer is D

Cheers,
Brent