Geometry and TrigonometryVery hardTriangle inequality
Geometry and Trigonometry practice question
Two sides of a triangle have lengths 7 and 10. If the third side has an integer length, what is the greatest possible length of the third side?
Answer: 16
In any triangle, each side is shorter than the sum of the other two. So the third side must be less than 7 + 10 = 17. The greatest integer less than 17 is 16. (It must also be greater than 10 − 7 = 3, which 16 satisfies.)
Common wrong answers
- If you answered 3
- 3 is the lower limit, 10 − 7, and the third side must be strictly greater than it. The question asks for the greatest possible length.
- If you answered 13
- 13 works, but it isn't the greatest. Any integer up to 16 keeps the third side below 17.
- If you answered 17
- 17 is the sum of the other two sides, but the third side must be strictly LESS than that sum — with 17 the 'triangle' would flatten into a straight line.
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