The number of possible triangles with any three of the lengths 1.2 cm, 4.2 cm, 5.9 cm and 8.1 cm is:

Options

A. Three
B. Two
C. One
D. Four

Explanation

To determine the number of possible triangles that can be formed with the lengths 1.2 cm, 4.2 cm, 5.9 cm, and 8.1 cm, we need to 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 all combinations of three lengths from the given set.

### Step 1: List all combinations of three lengths
The lengths provided are:
- 1.2 cm
- 4.2 cm
- 5.9 cm
- 8.1 cm

The combinations of three lengths are:
1. \( 1.2, 4.2, 5.9 \)
2. \( 1.2, 4.2, 8.1 \)
3. \( 1.2, 5.9, 8.1 \)
4. \( 4.2, 5.9, 8.1 \)

### Step 2: Check each combination against the triangle inequality

#### Combination 1: \( 1.2, 4.2, 5.9 \)
- Check \( 1.2 + 4.2 > 5.9 \):
- \( 5.4 > 5.9 \) (False)
- This combination does not form a triangle.

#### Combination 2: \( 1.2, 4.2, 8.1 \)
- Check \( 1.2 + 4.2 > 8.1 \):
- \( 5.4 > 8.1 \) (False)
- This combination does not form a triangle.

#### Combination 3: \( 1.2, 5.9, 8.1 \)
- Check \( 1.2 + 5.9 > 8.1 \):
- \( 7.1 > 8.1 \) (False)
- This combination does not form a triangle.

#### Combination 4: \( 4.2, 5.9, 8.1 \)
- Check \( 4.2 + 5.9 > 8.1 \):
- \( 10.1 > 8.1 \) (True)
- Check \( 4.2 + 8.1 > 5.9 \):
- \( 12.3 > 5.9 \) (True)
- Check \( 5.9 + 8.1 > 4.2 \):
- \( 14.0 > 4.2 \) (True)
- This combination forms a triangle.

### Conclusion
Out of the four combinations checked, only one combination \( (4.2, 5.9, 8.1) \) satisfies the triangle inequality conditions and can form a triangle.

Thus, the number of possible triangles that can be formed is **1**.