Which of the following measures of sides can be used to construct a triangle?

Options

A. 3 cm, 8 cm, 14 cm
B. 6 cm, 3 cm, 4 cm
C. 8 cm, 13 cm, 5 cm
D. 4 cm, 5 cm, 10 cm

Answer

To determine which set of side lengths can be used to construct a triangle, we will apply the triangle inequality theorem. This theorem states that for any triangle with sides of lengths a, b, and c, the following conditions must be satisfied:

1. a + b > c
2. a + c > b
3. b + c > a

We will check each set of side lengths to see if they satisfy these conditions.

### Step-by-Step Solution:

**Option 1: 3 cm, 8 cm, 14 cm**
1. Check if 3 + 8 > 14:
- 3 + 8 = 11, which is not greater than 14.
2. Since this condition fails, we cannot construct a triangle with these sides.

**Option 2: 6 cm, 3 cm, 4 cm**
1. Check if 6 + 3 > 4:
- 6 + 3 = 9, which is greater than 4.
2. Check if 6 + 4 > 3:
- 6 + 4 = 10, which is greater than 3.
3. Check if 3 + 4 > 6:
- 3 + 4 = 7, which is greater than 6.
4. All conditions are satisfied, so we can construct a triangle with these sides.

**Option 3: 8 cm, 13 cm, 5 cm**
1. Check if 8 + 13 > 5:
- 8 + 13 = 21, which is greater than 5.
2. Check if 8 + 5 > 13:
- 8 + 5 = 13, which is not greater than 13.
3. Since this condition fails, we cannot construct a triangle with these sides.

**Option 4: 4 cm, 5 cm, 10 cm**
1. Check if 4 + 5 > 10:
- 4 + 5 = 9, which is not greater than 10.
2. Since this condition fails, we cannot construct a triangle with these sides.

### Conclusion:
The only set of side lengths that can be used to construct a triangle is **Option 2: 6 cm, 3 cm, and 4 cm**.

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