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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, 9, comma, 13, comma, 22, right curly bracket{9,13,22}
left curly bracket, 13, comma, 16, comma, 26, right curly bracket{13,16,26}
left curly bracket, 9, comma, 11, comma, 22, right curly bracket{9,11,22}
left curly bracket, 7, comma, 12, comma, 20, right curly bracket{7,12,20}

1 answer

To determine which of the given sets of numbers can represent the sides of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides of lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

We'll check each of the provided sets of numbers against these inequalities.

  1. Set \({9, 13, 22}\):

    • \(9 + 13 = 22\) (not greater)
    • \(9 + 22 = 31 > 13\) (true)
    • \(13 + 22 = 35 > 9\) (true)
    • Not a triangle (fails the first condition)

  2. Set \({13, 16, 26}\):

    • \(13 + 16 = 29 > 26\) (true)
    • \(13 + 26 = 39 > 16\) (true)
    • \(16 + 26 = 42 > 13\) (true)
    • Is a triangle (all conditions are satisfied)

  3. Set \({9, 11, 22}\):

    • \(9 + 11 = 20 < 22\) (not greater)
    • \(9 + 22 = 31 > 11\) (true)
    • \(11 + 22 = 33 > 9\) (true)
    • Not a triangle (fails the first condition)

  4. Set \({7, 12, 20}\):

    • \(7 + 12 = 19 < 20\) (not greater)
    • \(7 + 20 = 27 > 12\) (true)
    • \(12 + 20 = 32 > 7\) (true)
    • Not a triangle (fails the first condition)

Based on the triangle inequality conditions, the only set that can represent the sides of a triangle is:

  • \({13, 16, 26}\).