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which set of side lengths will form a triangle? all lengths are given in meters.

60 26 33.

26 12 12.

50 30 80.

50 60 26.

1 answer

To determine if a set of side lengths can form a triangle, we can use the triangle inequality theorem. The theorem states that for three lengths \(a\), \(b\), and \(c\), the following conditions must be satisfied:

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

Let's apply this rule to each set of side lengths.

  1. Lengths: 60, 26, 33

    • \(60 + 26 > 33 \) → \(86 > 33\) (True)
    • \(60 + 33 > 26 \) → \(93 > 26\) (True)
    • \(26 + 33 > 60 \) → \(59 > 60\) (False)

    This set does NOT form a triangle.

  2. Lengths: 26, 12, 12

    • \(26 + 12 > 12 \) → \(38 > 12\) (True)
    • \(26 + 12 > 12 \) → \(38 > 12\) (True)
    • \(12 + 12 > 26 \) → \(24 > 26\) (False)

    This set does NOT form a triangle.

  3. Lengths: 50, 30, 80

    • \(50 + 30 > 80 \) → \(80 > 80\) (False)
    • \(50 + 80 > 30 \) → \(130 > 30\) (True)
    • \(30 + 80 > 50 \) → \(110 > 50\) (True)

    This set does NOT form a triangle.

  4. Lengths: 50, 60, 26

    • \(50 + 60 > 26 \) → \(110 > 26\) (True)
    • \(50 + 26 > 60 \) → \(76 > 60\) (True)
    • \(60 + 26 > 50 \) → \(86 > 50\) (True)

    This set DOES form a triangle.

Summary:

  • The side lengths 50, 60, 26 can form a triangle.
  • The other sets of lengths cannot form a triangle.