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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, 12, comma, 15, comma, 28, right curly bracket{12,15,28}
left curly bracket, 11, comma, 21, comma, 32, right curly bracket{11,21,32}
left curly bracket, 12, comma, 22, comma, 33, right curly bracket{12,22,33}
left curly bracket, 12, comma, 24, comma, 36, right curly bracket{12,24,36}

1 answer

To determine whether a set of numbers can represent the sides of a triangle, we can apply the triangle inequality theorem. This theorem states that for any three sides \( a \), \( b \), and \( c \) (where \( a \leq b \leq c \)), the following conditions must hold:

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

However, the most commonly used inequality to check is \( a + b > c \) when the sides are ordered.

Let's analyze each of the given sets:

  1. Set {12, 15, 28}

    • Ordered: \( 12, 15, 28 \)
    • Check: \( 12 + 15 = 27 \) which is not greater than 28.
    • This set cannot represent the sides of a triangle.

  2. Set {11, 21, 32}

    • Ordered: \( 11, 21, 32 \)
    • Check: \( 11 + 21 = 32 \) which is not greater than 32.
    • This set cannot represent the sides of a triangle.

  3. Set {12, 22, 33}

    • Ordered: \( 12, 22, 33 \)
    • Check: \( 12 + 22 = 34 \) which is greater than 33.
    • Check: \( 12 + 33 = 45 \) which is greater than 22.
    • Check: \( 22 + 33 = 55 \) which is greater than 12.
    • This set can represent the sides of a triangle.

  4. Set {12, 24, 36}

    • Ordered: \( 12, 24, 36 \)
    • Check: \( 12 + 24 = 36 \) which is not greater than 36.
    • This set cannot represent the sides of a triangle.

Thus, the only set that can represent the sides of a triangle is {12, 22, 33}.