If the three side lengths of a right triangle are consecutive integers x−1, x, x+1, determine the hypotenuse.
Aptitude
Plane Geometry
Difficulty: Easy
Choose an option
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A5
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B4
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C1
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D0
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ENone of these
Answer
Correct Answer: 5
Explanation
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:
(x+1)^2 = x^2 + (x−1)^2x^2 + 2x + 1 = x^2 + x^2 − 2x + 1Simplify ⇒ 0 = x^2 − 4x ⇒ x(x − 4) = 0x = 4 (reject x = 0 as a length)Hypotenuse = x + 1 = 5Verification / 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