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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, 14, comma, 25, comma, 39, right curly bracket{14,25,39}
left curly bracket, 14, comma, 16, comma, 30, right curly bracket{14,16,30}
left curly bracket, 14, comma, 27, comma, 40, right curly bracket{14,27,40}
left curly bracket, 11, comma, 24, comma, 36, right curly bracket{11,24,36}

1 answer

To determine if three lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that the sum of the lengths of any two sides must be greater than the length of the third side.

Let’s evaluate each set of numbers:

  1. Set {14, 25, 39}:

    • \(14 + 25 = 39\) (not greater than 39)
    • \(14 + 39 = 53\) (greater than 25)
    • \(25 + 39 = 64\) (greater than 14)
    • Fails the triangle inequality.

  2. Set {14, 16, 30}:

    • \(14 + 16 = 30\) (not greater than 30)
    • \(14 + 30 = 44\) (greater than 16)
    • \(16 + 30 = 46\) (greater than 14)
    • Fails the triangle inequality.

  3. Set {14, 27, 40}:

    • \(14 + 27 = 41\) (greater than 40)
    • \(14 + 40 = 54\) (greater than 27)
    • \(27 + 40 = 67\) (greater than 14)
    • Passes the triangle inequality.

  4. Set {11, 24, 36}:

    • \(11 + 24 = 35\) (not greater than 36)
    • \(11 + 36 = 47\) (greater than 24)
    • \(24 + 36 = 60\) (greater than 11)
    • Fails the triangle inequality.

Based on this analysis, the only set that satisfies the Triangle Inequality Theorem is:

Set {14, 27, 40}.