Question

Which set of side lengths will form a triangle? All measurements are in feet.
8, 6, 2
3, 3, 6
4, 1, 6
5, 2, 4

Answer 1

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

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

Let's apply these conditions to each set of side lengths:

1. **For the sides 8, 6, 2:**
- \(8 + 6 = 14 > 2\) ✔️
- \(8 + 2 = 10 > 6\) ✔️
- \(6 + 2 = 8 \not> 8\) ✖️
This set does not form a triangle.

2. **For the sides 3, 3, 6:**
- \(3 + 3 = 6 \not> 6\) ✖️
- \(3 + 6 = 9 > 3\) ✔️
- \(3 + 6 = 9 > 3\) ✔️
This set does not form a triangle.

3. **For the sides 4, 1, 6:**
- \(4 + 1 = 5 \not> 6\) ✖️
- \(4 + 6 = 10 > 1\) ✔️
- \(1 + 6 = 7 > 4\) ✔️
This set does not form a triangle.

4. **For the sides 5, 2, 4:**
- \(5 + 2 = 7 > 4\) ✔️
- \(5 + 4 = 9 > 2\) ✔️
- \(2 + 4 = 6 > 5\) ✔️
This set forms a triangle.

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