Question

Which set of side lengths will form a triangle? All measurements are in feet.

A. 4,1,6
B. 8,6,2
C. 3,3,6
D. 5,2,4

Answer 1

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

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

We will check each set of side lengths against these conditions.

A. **4, 1, 6**
- \( 4 + 1 = 5 \) (not greater than 6)
- \( 4 + 6 = 10 > 1 \)
- \( 1 + 6 = 7 > 4 \)
- **Fails (not a triangle)**

B. **8, 6, 2**
- \( 8 + 6 = 14 > 2 \)
- \( 8 + 2 = 10 > 6 \)
- \( 6 + 2 = 8 \) (not greater than 8)
- **Fails (not a triangle)**

C. **3, 3, 6**
- \( 3 + 3 = 6 \) (not greater than 6)
- \( 3 + 6 = 9 > 3 \)
- \( 3 + 6 = 9 > 3 \)
- **Fails (not a triangle)**

D. **5, 2, 4**
- \( 5 + 2 = 7 > 4 \)
- \( 5 + 4 = 9 > 2 \)
- \( 2 + 4 = 6 > 5 \)
- **Passes (this is a triangle)**

The only set of side lengths that can form a triangle is **D. 5, 2, 4**.