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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, 11, comma, 17, comma, 26, right curly bracket{11,17,26}
left curly bracket, 10, comma, 14, comma, 25, right curly bracket{10,14,25}
left curly bracket, 14, comma, 26, comma, 38, right curly bracket{14,26,38}
left curly bracket, 11, comma, 13, comma, 23, right curly bracket{11,13,23}

1 answer

To determine which set of numbers could not represent the sides of a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with sides \(a\), \(b\), and \(c\), the following conditions must hold true:

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

Let's verify each set of numbers:

  1. Set \({11, 17, 26}\):

    • \(11 + 17 = 28 > 26\) (true)
    • \(11 + 26 = 37 > 17\) (true)
    • \(17 + 26 = 43 > 11\) (true)
    • Conclusion: This set can form a triangle.

  2. Set \({10, 14, 25}\):

    • \(10 + 14 = 24\) (not greater than \(25\))
    • Therefore, this does not satisfy the triangle inequality.
    • Conclusion: This set cannot form a triangle.

  3. Set \({14, 26, 38}\):

    • \(14 + 26 = 40 > 38\) (true)
    • \(14 + 38 = 52 > 26\) (true)
    • \(26 + 38 = 64 > 14\) (true)
    • Conclusion: This set can form a triangle.

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

    • \(11 + 13 = 24 > 23\) (true)
    • \(11 + 23 = 34 > 13\) (true)
    • \(13 + 23 = 36 > 11\) (true)
    • Conclusion: This set can form a triangle.

From the analysis, the set of numbers that cannot represent the three sides of a triangle is:

\({10, 14, 25}\).