Question

Which of the following combinations of side lengths would form a​ triangle?
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Part 1
Select all that apply.
A.
12 ​in., 9.25 ​in., 20 in.
B.
4 ​ft, 11 ​ft, 6 ft
C.
3.25 ​ft, 4 ​ft, 5 ft
D.
4 ​m, 4 ​m, 5 m
E.
14 ​in., 7 ​in., 4.5 in.
F.
11 ​yd, 6 ​yd, 3 yd

Answer 1

To determine whether a set of three 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 \( c \) (where \( a \leq b \leq c \)), the following must hold:

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

We can check each combination of side lengths one by one:

**A. 12 in., 9.25 in., 20 in.**
- \( 12 + 9.25 = 21.25 > 20 \) (True)
- \( 12 + 20 = 32 > 9.25 \) (True)
- \( 9.25 + 20 = 29.25 > 12 \) (True)
This combination forms a triangle.

**B. 4 ft, 11 ft, 6 ft**
- \( 4 + 6 = 10 < 11 \) (False)
This combination does not form a triangle.

**C. 3.25 ft, 4 ft, 5 ft**
- \( 3.25 + 4 = 7.25 > 5 \) (True)
- \( 3.25 + 5 = 8.25 > 4 \) (True)
- \( 4 + 5 = 9 > 3.25 \) (True)
This combination forms a triangle.

**D. 4 m, 4 m, 5 m**
- \( 4 + 4 = 8 > 5 \) (True)
- \( 4 + 5 = 9 > 4 \) (True)
- \( 4 + 5 = 9 > 4 \) (True)
This combination forms a triangle.

**E. 14 in., 7 in., 4.5 in.**
- \( 14 + 7 = 21 > 4.5 \) (True)
- \( 14 + 4.5 = 18.5 > 7 \) (True)
- \( 7 + 4.5 = 11.5 < 14 \) (False)
This combination does not form a triangle.

**F. 11 yd, 6 yd, 3 yd**
- \( 11 + 6 = 17 > 3 \) (True)
- \( 11 + 3 = 14 > 6 \) (True)
- \( 6 + 3 = 9 < 11 \) (False)
This combination does not form a triangle.

**Final results:**
The combinations that form a triangle are:

- **A. 12 in., 9.25 in., 20 in.**
- **C. 3.25 ft, 4 ft, 5 ft**
- **D. 4 m, 4 m, 5 m**