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

5,9,14
510,15
8,10,16
3,15,17

1 answer

To determine which set of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\):

  1. \(a + b > c\)
  2. \(a + c > b\)
  3. \(b + c > a\)

We will apply this theorem to each set of lengths provided.

  1. 5, 9, 14:

    • \(5 + 9 = 14\) (not greater)
    • \(5 + 14 = 19 > 9\)
    • \(9 + 14 = 23 > 5\)
    • Conclusion: This set cannot form a triangle.
  2. 510, 15:

    • This set does not have three lengths (only two provided).
    • Conclusion: This set does not represent a triangle.
  3. 8, 10, 16:

    • \(8 + 10 = 18 > 16\)
    • \(8 + 16 = 24 > 10\)
    • \(10 + 16 = 26 > 8\)
    • Conclusion: This set can form a triangle.
  4. 3, 15, 17:

    • \(3 + 15 = 18 > 17\)
    • \(3 + 17 = 20 > 15\)
    • \(15 + 17 = 32 > 3\)
    • Conclusion: This set can form a triangle.

Final Results: The side lengths that can form a triangle are:

  • 8, 10, 16
  • 3, 15, 17