Problem 10 · AMC 8 Stretch
Core
Geometry & Measurement
Number Theory
intelligent-guessing-and-testingaccount-for-all-possibilities
Using whole-number inch marks, what is the SHORTEST wire that can be bent into a RIGHT triangle? And what is the shortest wire that can be bent into an OBTUSE (one angle bigger than \(90^\circ\)) triangle?
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Answer: 12 inches (right, 3-4-5) and 7 inches (obtuse, 2-2-3)
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Hint 1 of 4
A whole-number right triangle must be a Pythagorean triple (sides where \(a^2 + b^2 = c^2\)). What is the smallest famous one?
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Hint 2 of 4
The smallest Pythagorean triple is 3-4-5. Add the sides to get the wire length.
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Hint 3 of 4
For obtuse, you need a real triangle (\(a + b > c\)) where the longest side is 'too long': \(a^2 + b^2 < c^2\). Try short triples and test.
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Approach: Smallest Pythagorean triple, then test short triples for obtuse
- A whole-number right triangle is a Pythagorean triple \(a^2+b^2=c^2\). The smallest is 3-4-5, perimeter \(3+4+5 = 12\), and no shorter whole-number triangle is right. So the shortest right-triangle wire is 12 inches.
- For obtuse we want a real triangle whose longest side satisfies \(a^2 + b^2 < c^2\). Check perimeters in order: at perimeter 6 or less the only triangles (1-1-1, 1-2-2, 2-2-2) are equilateral or acute (1-2-2: \(1+4=5 > 4\), acute).
- Perimeter 7: try 2-2-3. Since \(2^2 + 2^2 = 8 < 9 = 3^2\), the angle across from the side of 3 is obtuse.
- So the shortest obtuse-triangle wire is 7 inches (2-2-3). Answer: right = 12 inches, obtuse = 7 inches.
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