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Which of the following sets of numbers could represent the three sides of a triangle?

Answer
Multiple Choice Answers
left curly bracket, 11, comma, 22, comma, 32, right curly bracket{11,22,32}
left curly bracket, 8, comma, 22, comma, 31, right curly bracket{8,22,31}
left curly bracket, 13, comma, 17, comma, 30, right curly bracket{13,17,30}
left curly bracket, 10, comma, 23, comma, 33, right curly bracket{10,23,33}

1 answer

To determine if a set of three numbers can represent the sides of a triangle, we can apply the triangle inequality theorem. According to this theorem, for any three sides \(a\), \(b\), and \(c\) of a triangle, the following conditions must all hold true:

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

Let's evaluate each of the provided sets of numbers:

  1. Set {11, 22, 32}

    • \(11 + 22 = 33 > 32\) (True)
    • \(11 + 32 = 43 > 22\) (True)
    • \(22 + 32 = 54 > 11\) (True)
      All conditions are satisfied.

  2. Set {8, 22, 31}

    • \(8 + 22 = 30 < 31\) (False)
    • \(8 + 31 = 39 > 22\) (True)
    • \(22 + 31 = 53 > 8\) (True)
      Not all conditions are satisfied.

  3. Set {13, 17, 30}

    • \(13 + 17 = 30\) (Not greater than, but equal to)
    • \(13 + 30 = 43 > 17\) (True)
    • \(17 + 30 = 47 > 13\) (True)
      Not all conditions are satisfied.

  4. Set {10, 23, 33}

    • \(10 + 23 = 33\) (Not greater than, but equal to)
    • \(10 + 33 = 43 > 23\) (True)
    • \(23 + 33 = 56 > 10\) (True)
      Not all conditions are satisfied.

Based on our evaluation, the only set that satisfies the triangle inequality theorem is:

Set {11, 22, 32}.