Geometry and TrigonometryHardTriangle inequality at the boundary
Geometry and Trigonometry practice question
Two sides of a triangle have lengths 5 and 12. Which of the following could NOT be the length of the third side?
- A. 17Correct
- B. 8
- C. 10
- D. 15
Answer: A. 17
The third side must be strictly between 12 − 5 = 7 and 12 + 5 = 17. A side of exactly 17 would make 5 + 12 = 17, and the 'triangle' would collapse into a straight segment. So 17 is impossible.
Why the other answers are wrong
- B. 8
- 8 works: 5 + 8 = 13 > 12, and it is greater than 12 − 5 = 7.
- C. 10
- 10 works: 5 + 10 = 15 > 12.
- D. 15
- 15 works: it is less than 5 + 12 = 17 and greater than 7.
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