Which set of side lengths does not form a triangle? All lengths are given in inches.

• 17, 11, 9
• 19, 11, 9
• 18, 11, 8
• 18, 10, 8

1 answer

To determine whether a set of side lengths can form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for any three lengths \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:

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

We will check each set of side lengths:

  1. 17, 11, 9

    • \(17 + 11 = 28 > 9\) (True)
    • \(17 + 9 = 26 > 11\) (True)
    • \(11 + 9 = 20 > 17\) (True)
    • This set can form a triangle.
  2. 19, 11, 9

    • \(19 + 11 = 30 > 9\) (True)
    • \(19 + 9 = 28 > 11\) (True)
    • \(11 + 9 = 20 > 19\) (False)
    • This set cannot form a triangle.
  3. 18, 11, 8

    • \(18 + 11 = 29 > 8\) (True)
    • \(18 + 8 = 26 > 11\) (True)
    • \(11 + 8 = 19 > 18\) (True)
    • This set can form a triangle.
  4. 18, 10, 8

    • \(18 + 10 = 28 > 8\) (True)
    • \(18 + 8 = 26 > 10\) (True)
    • \(10 + 8 = 18 = 18\) (False, equality does not satisfy the triangle inequality)
    • This set cannot form a triangle.

From the checks above, the sets that do not form a triangle are 19, 11, 9 and 18, 10, 8.