Question

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 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

Answer 1

To determine if a set of lengths can form a triangle, we can use the Triangle Inequality Theorem. This theorem states that for three side lengths \(a\), \(b\), and \(c\), the following conditions must hold true:

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

Let’s check each option using these conditions.

### Option 1: Side lengths 3, 3, 6
1. \(3 + 3 > 6\) → \(6 > 6\) (False)
2. \(3 + 6 > 3\) → \(9 > 3\) (True)
3. \(3 + 6 > 3\) → \(9 > 3\) (True)

Since the first condition is false, **these lengths cannot form a triangle**.

### Option 2: Side lengths 6, 8, 10
1. \(6 + 8 > 10\) → \(14 > 10\) (True)
2. \(6 + 10 > 8\) → \(16 > 8\) (True)
3. \(8 + 10 > 6\) → \(18 > 6\) (True)

All conditions are true, so **these lengths can form a triangle**.

### Option 3: Side lengths 4, 6, 13
1. \(4 + 6 > 13\) → \(10 > 13\) (False)
2. \(4 + 13 > 6\) → \(17 > 6\) (True)
3. \(6 + 13 > 4\) → \(19 > 4\) (True)

Since the first condition is false, **these lengths cannot form a triangle**.

### Summary
- **Option 1 (3, 3, 6)**: Cannot form a triangle.
- **Option 2 (6, 8, 10)**: Can form a triangle.
- **Option 3 (4, 6, 13)**: Cannot form a triangle.

Therefore, **option 2** is the only set that can be used to form a triangle.