Problem 17 · 1992 AJHSME
Hard
Geometry & Measurement
triangle-inequality
The sides of a triangle have lengths 6.5, 10, and s, where s is a whole number. What is the smallest possible value of s?
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Answer: B — 4.
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Hint 1 of 3
Picture the two fixed sides (6.5 and 10) hinged together. If the third side is too short, the two free ends can't reach each other to close the triangle. What's the shortest the third side can be and still connect them?
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Hint 2 of 3
The triangle inequality: any side must be longer than the difference of the other two (and shorter than their sum). The "smallest" question is governed by the difference.
Still stuck? Show hint 3 →
Hint 3 of 3
10 − 6.5 = 3.5 is the floor — s must be strictly more than 3.5, and it's a whole number.
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Approach: the short side must beat the gap between the other two
- Imagine the sides 6.5 and 10 pinned at one end. To swing their far ends together and close a triangle, the third side must bridge at least the difference 10 − 6.5 = 3.5. If s were 3.5 or less, the triangle flattens and won't form.
- So s must be greater than 3.5. The smallest whole number bigger than 3.5 is 4.
- Why this transfers: for three lengths to make a real (non-flat) triangle, each side must be less than the sum AND more than the difference of the other two. When a problem asks for the smallest side, it's the difference rule that bites; for the largest, it's the sum rule.
- Trap: 3 fails (3 + 6.5 = 9.5 < 10, can't reach), confirming 4 is the first that works.
Mark:
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