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Which of the following sets of numbers could not represent the three sides of a triangle?

Answer
Multiple Choice Answers
left curly bracket, 4, comma, 16, comma, 20, right curly bracket{4,16,20}
left curly bracket, 11, comma, 13, comma, 23, right curly bracket{11,13,23}
left curly bracket, 8, comma, 10, comma, 16, right curly bracket{8,10,16}
left curly bracket, 8, comma, 11, comma, 17, right curly bracket{8,11,17}

1 answer

To determine which set of numbers cannot represent the sides of a triangle, we can use the triangle inequality theorem. The theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following inequalities must be satisfied:

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

We can check each set of numbers based on these inequalities:

  1. Set \({4, 16, 20}\):

    • \(4 + 16 = 20\) (not greater than)
    • \(4 + 20 = 24 > 16\) (satisfied)
    • \(16 + 20 = 36 > 4\) (satisfied)

    Since \(4 + 16\) is not greater than \(20\), this set does not satisfy the triangle inequality.

  2. Set \({11, 13, 23}\):

    • \(11 + 13 = 24 > 23\) (satisfied)
    • \(11 + 23 = 34 > 13\) (satisfied)
    • \(13 + 23 = 36 > 11\) (satisfied)

    This set satisfies all inequalities.

  3. Set \({8, 10, 16}\):

    • \(8 + 10 = 18 > 16\) (satisfied)
    • \(8 + 16 = 24 > 10\) (satisfied)
    • \(10 + 16 = 26 > 8\) (satisfied)

    This set satisfies all inequalities.

  4. Set \({8, 11, 17}\):

    • \(8 + 11 = 19 > 17\) (satisfied)
    • \(8 + 17 = 25 > 11\) (satisfied)
    • \(11 + 17 = 28 > 8\) (satisfied)

    This set satisfies all inequalities.

Based on the checks above, the set that cannot represent the sides of a triangle is \({4, 16, 20}\).