If x
2
−1, 2x and x
2
+1 are the three sides of a right-angled triangle, then which of the following can be its hypotenuse?

In a right-angled triangle, the hypotenuse is always the longest side.
Let's analyze the three given sides: x
2
−1, 2x, and x
2
+1.

For any real value of x>1:

x
2
+1>x
2
−1
Also, let's verify using Pythagoras' theorem whether these sides form a Pythagorean triplet where x
2
+1 is the hypotenuse:

Base
2
+Perpendicular
2
=(x
2
−1)
2
+(2x)
2

=(x
4
−2x
2
+1)+4x
2

=x
4
+2x
2
+1
=(x
2
+1)
2
=Hypotenuse
2

Since the relation holds true, (x
2
+1) is the largest side and represents the hypotenuse of the triangle.