Taking any three of the line segments out of segments of length 2 cm, 3 cm, 5 cm and 6 cm, the number of triangles that can be formed is :

Options

A. 3
B. 2
C. 1
D. 4

Answer

To determine how many triangles can be formed using any three of the given line segments of lengths 2 cm, 3 cm, 5 cm, and 6 cm, we can follow these steps:

### Step 1: Understand the Triangle Inequality Theorem
The triangle inequality theorem states that for any three sides \(a\), \(b\), and \(c\) to form a triangle, the following conditions must be satisfied:
1. \(a + b > c\)
2. \(a + c > b\)
3. \(b + c > a\)

### Step 2: List All Combinations of Three Segments
We need to find all combinations of three segments from the given lengths. The combinations are:
1. (2 cm, 3 cm, 5 cm)
2. (2 cm, 3 cm, 6 cm)
3. (2 cm, 5 cm, 6 cm)
4. (3 cm, 5 cm, 6 cm)

### Step 3: Check Each Combination Against the Triangle Inequality
Now, we will check each combination to see if they satisfy the triangle inequality.

1. **Combination (2 cm, 3 cm, 5 cm)**:
- \(2 + 3 = 5\) (not greater than 5)
- This combination does not form a triangle.

2. **Combination (2 cm, 3 cm, 6 cm)**:
- \(2 + 3 = 5\) (not greater than 6)
- This combination does not form a triangle.

3. **Combination (2 cm, 5 cm, 6 cm)**:
- \(2 + 5 = 7 > 6\)
- \(2 + 6 = 8 > 5\)
- \(5 + 6 = 11 > 2\)
- This combination forms a triangle.

4. **Combination (3 cm, 5 cm, 6 cm)**:
- \(3 + 5 = 8 > 6\)
- \(3 + 6 = 9 > 5\)
- \(5 + 6 = 11 > 3\)
- This combination forms a triangle.

### Step 4: Count the Valid Combinations
From our checks, we found that only the combinations (2 cm, 5 cm, 6 cm) and (3 cm, 5 cm, 6 cm) can form triangles. Thus, the total number of triangles that can be formed is **2**.

### Final Answer
The number of triangles that can be formed is **2**.

---