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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, 23, comma, 35, right curly bracket{11,23,35}
left curly bracket, 7, comma, 17, comma, 23, right curly bracket{7,17,23}
left curly bracket, 8, comma, 22, comma, 32, right curly bracket{8,22,32}
left curly bracket, 9, comma, 18, comma, 27, right curly bracket{9,18,27}

1 answer

To determine which of the given sets of numbers can represent the sides of a triangle, we need to use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\) (where \(a\), \(b\), and \(c\) are the lengths of the sides), the following inequalities must hold true:

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

Let's evaluate each set of numbers:

  1. Set {11, 23, 35}:

    • \(11 + 23 = 34 > 35\) (False)
    • \(11 + 35 = 46 > 23\) (True)
    • \(23 + 35 = 58 > 11\) (True)
    • This set does not form a triangle.

  2. Set {7, 17, 23}:

    • \(7 + 17 = 24 > 23\) (True)
    • \(7 + 23 = 30 > 17\) (True)
    • \(17 + 23 = 40 > 7\) (True)
    • This set forms a triangle.

  3. Set {8, 22, 32}:

    • \(8 + 22 = 30 > 32\) (False)
    • \(8 + 32 = 40 > 22\) (True)
    • \(22 + 32 = 54 > 8\) (True)
    • This set does not form a triangle.

  4. Set {9, 18, 27}:

    • \(9 + 18 = 27\) (Not greater than)
    • \(9 + 27 = 36 > 18\) (True)
    • \(18 + 27 = 45 > 9\) (True)
    • This set does not form a triangle.

Based on the evaluations:

  • The only set that can represent the sides of a triangle is {7, 17, 23}.