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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, 13, comma, 25, comma, 40, right curly bracket{13,25,40}
left curly bracket, 10, comma, 16, comma, 25, right curly bracket{10,16,25}
left curly bracket, 6, comma, 18, comma, 25, right curly bracket{6,18,25}
left curly bracket, 13, comma, 22, comma, 35, right curly bracket{13,22,35}

1 answer

To determine if a set of numbers can represent the lengths of 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 inequalities must hold:

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

Let’s evaluate each set of numbers:

  1. Set: {13, 25, 40}

    • \(13 + 25 = 38\) (not greater than 40)
    • \(13 + 40 = 53\) (greater than 25)
    • \(25 + 40 = 65\) (greater than 13)
    • This set does not satisfy the triangle inequalities.

  2. Set: {10, 16, 25}

    • \(10 + 16 = 26\) (greater than 25)
    • \(10 + 25 = 35\) (greater than 16)
    • \(16 + 25 = 41\) (greater than 10)
    • This set satisfies all triangle inequalities.

  3. Set: {6, 18, 25}

    • \(6 + 18 = 24\) (not greater than 25)
    • \(6 + 25 = 31\) (greater than 18)
    • \(18 + 25 = 43\) (greater than 6)
    • This set does not satisfy the triangle inequalities.

  4. Set: {13, 22, 35}

    • \(13 + 22 = 35\) (not greater than 35)
    • \(13 + 35 = 48\) (greater than 22)
    • \(22 + 35 = 57\) (greater than 13)
    • This set does not satisfy the triangle inequalities.

Based on the evaluations, the only set of numbers that could represent the lengths of the sides of a triangle is:

{10, 16, 25}.