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The following side lengths, in meters, were given to a carpenter to build a front porch with a triangular design. The carpenter needs to determine which set of lengths will make a triangle to be able to use it in his design. Option 1: Side lengths: 3, 3, 6 Option 2: Side lengths: 6, 8, 10 Option 3: Side lengths: 4, 6, 13 (1 point) Responses Option 1 Option 1 Option 2 Option 2 Option 3

1 answer

To determine whether a set of side lengths can form a triangle, we can use the triangle inequality theorem. This theorem states that for any triangle with side lengths \(a\), \(b\), and \(c\):

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

Let's apply this to each of the options.

Option 1: Side lengths 3, 3, 6

  • Check: \(3 + 3 > 6\) → \(6 > 6\) (not true)
  • Check: \(3 + 6 > 3\) → \(9 > 3\) (true)
  • Check: \(3 + 6 > 3\) → \(9 > 3\) (true)

Since the first condition is not met, these lengths do not form a triangle.

Option 2: Side lengths 6, 8, 10

  • Check: \(6 + 8 > 10\) → \(14 > 10\) (true)
  • Check: \(6 + 10 > 8\) → \(16 > 8\) (true)
  • Check: \(8 + 10 > 6\) → \(18 > 6\) (true)

All conditions are met, so these lengths do form a triangle.

Option 3: Side lengths 4, 6, 13

  • Check: \(4 + 6 > 13\) → \(10 > 13\) (not true)
  • Check: \(4 + 13 > 6\) → \(17 > 6\) (true)
  • Check: \(6 + 13 > 4\) → \(19 > 4\) (true)

Since the first condition is not met, these lengths do not form a triangle.

Conclusion:

The only option that can form a triangle is Option 2: Side lengths 6, 8, 10.