Three sides of a triangle are b2 − 1, b2 + 1 and 2b respectively. Then the…

Concept: In a right-angled triangle, the square of the longest side (the hypotenuse) equals the sum of the squares of the other two sides — the Pythagorean theorem. If three given sides satisfy this relation, the triangle is right-angled, and the right angle — always the greatest angle of any triangle — lies opposite the longest side.

Application:

1. The three given sides are b2 − 1, b2 + 1 and 2b. For b > 1, compare the three expressions: b2 + 1 is greater than b2 − 1, and also greater than 2b (since (b − 1)2 ≥ 0 gives b2 + 1 ≥ 2b). So b2 + 1 is the longest side.

2. Test the Pythagorean relation with b2 + 1 as the hypotenuse: (b2 + 1)2 should equal (b2 − 1)2 + (2b)2.

3. Expand the left side: (b2 + 1)2 = b4 + 2b2 + 1.

4. Expand the right side: (b2 − 1)2 + (2b)2 = (b4 − 2b2 + 1) + 4b2 = b4 + 2b2 + 1.

5. Both sides are equal — b4 + 2b2 + 1 = b4 + 2b2 + 1 — for every value of b, so the relation holds identically.

6. Since the square of the longest side equals the sum of squares of the other two, the triangle is right-angled, with the right angle opposite the side b2 + 1 — and that right angle is the greatest angle of the triangle.

Cross-check: at b = 2, the sides become 3, 5 and 4 — the familiar 3-4-5 right triangle, whose greatest angle is indeed 90°, confirming the algebraic result.

So the greatest angle of the triangle is 90°.