Question

Which set of side lengths will form a triangle? All measurements are in feet.
(1 point)
0 5,2,4
0 4,1,6
0 8,6,2
0

Answer 1

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

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

Now, let's evaluate the provided sets of side lengths:

1. **For the sides 5, 2, and 4**:
- \(5 + 2 > 4\) → \(7 > 4\) (true)
- \(5 + 4 > 2\) → \(9 > 2\) (true)
- \(2 + 4 > 5\) → \(6 > 5\) (true)

Since all conditions are satisfied, these lengths can form a triangle.

2. **For the sides 4, 1, and 6**:
- \(4 + 1 > 6\) → \(5 > 6\) (false)
- \(4 + 6 > 1\) → \(10 > 1\) (true)
- \(1 + 6 > 4\) → \(7 > 4\) (true)

Since the first condition is not satisfied, these lengths cannot form a triangle.

3. **For the sides 8, 6, and 2**:
- \(8 + 6 > 2\) → \(14 > 2\) (true)
- \(8 + 2 > 6\) → \(10 > 6\) (true)
- \(6 + 2 > 8\) → \(8 > 8\) (false)

Since the third condition is not satisfied, these lengths cannot form a triangle.

In conclusion, the only set of side lengths that can form a triangle is **5, 2, and 4**.