Question

he 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. (Hint: Use the Triangle Inequality Theorem) 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

Answer 1

To determine whether each set of side lengths can form a triangle, we can use the Triangle Inequality Theorem, which states that for any three side lengths \(a\), \(b\), and \(c\), the following must be true:

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

Let's evaluate each option:

**Option 1: Side lengths: 3, 3, 6**
- \(3 + 3 > 6 \) (6 is not greater than 6, so this fails)
- \(3 + 6 > 3\) (9 > 3, this holds true)
- \(3 + 6 > 3\) (9 > 3, this holds true)

Since one of the inequalities does not hold, these lengths **do not form a triangle**.

**Option 2: Side lengths: 6, 8, 10**
- \(6 + 8 > 10\) (14 > 10, this holds true)
- \(6 + 10 > 8\) (16 > 8, this holds true)
- \(8 + 10 > 6\) (18 > 6, this holds true)

All inequalities hold, so these lengths **do form a triangle**.

**Option 3: Side lengths: 4, 6, 13**
- \(4 + 6 > 13\) (10 is not greater than 13, this fails)
- \(4 + 13 > 6\) (17 > 6, this holds true)
- \(6 + 13 > 4\) (19 > 4, this holds true)

Since one of the inequalities does not hold, these lengths **do not form a triangle**.

### Summary:
- Option 1: No triangle
- Option 2: Forms a triangle
- Option 3: No triangle

Therefore, **Option 2** is the only set of lengths that can form a triangle for the carpenter's design.