If the three side lengths of a right triangle are consecutive integers x−1, x, x+1, determine the hypotenuse.

Introduction / Context:We are told a right triangle has side lengths that are consecutive integers. The largest of the three must be the hypotenuse. We enforce the Pythagorean relation to determine the integer value(s).

Given Data / Assumptions:

• Sides are x−1, x, x+1 (consecutive integers).
• Right angle between the two smaller sides; hypotenuse = x+1.
• x > 1 (positive lengths).

Concept / Approach:

• Use (x+1)^2 = x^2 + (x−1)^2 and solve for x.
• Once x is known, the hypotenuse is x+1.

Step-by-Step Solution:

Verification / Alternative check:Check triple (3, 4, 5): 3^2 + 4^2 = 9 + 16 = 25 = 5^2; valid Pythagorean triple.

Why Other Options Are Wrong:

• 4, 1, 0 are not hypotenuse values for a positive right triangle in this context.
• None of these: Not applicable since 5 is exact.

Common Pitfalls:

• Misidentifying which side is the hypotenuse.

Final Answer:5