Question

Which set of side length can form a triangle? All lengths are given in inches.

a. 9, 17, 11
b. 12, 9, 3
c. 8, 18, 10
d. 5, 12, 6

Answer 1

To determine whether a set of side lengths can form a triangle, we can use the triangle inequality theorem. The triangle inequality states that for any three sides, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let's check each set:

a. **9, 17, 11**
- \(9 + 11 > 17\) → \(20 > 17\) (True)
- \(9 + 17 > 11\) → \(26 > 11\) (True)
- \(11 + 17 > 9\) → \(28 > 9\) (True)
**This set can form a triangle.**

b. **12, 9, 3**
- \(12 + 9 > 3\) → \(21 > 3\) (True)
- \(12 + 3 > 9\) → \(15 > 9\) (True)
- \(9 + 3 > 12\) → \(12 > 12\) (False)
**This set cannot form a triangle.**

c. **8, 18, 10**
- \(8 + 10 > 18\) → \(18 > 18\) (False)
- \(8 + 18 > 10\) → \(26 > 10\) (True)
- \(10 + 18 > 8\) → \(28 > 8\) (True)
**This set cannot form a triangle.**

d. **5, 12, 6**
- \(5 + 12 > 6\) → \(17 > 6\) (True)
- \(5 + 6 > 12\) → \(11 > 12\) (False)
- \(12 + 6 > 5\) → \(18 > 5\) (True)
**This set cannot form a triangle.**

In conclusion, only set **a. 9, 17, 11** can form a triangle.