Can we construct a triangle ABC in which `AB=3` cm, `BC=4` cm and `AC=8` cm ?

Answer

To determine if we can construct a triangle ABC with the given side lengths \( AB = 3 \) cm, \( BC = 4 \) cm, and \( AC = 8 \) cm, we can use the triangle inequality theorem. This theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

### Step-by-Step Solution:

1. **Identify the sides of the triangle**:
- Let \( AB = 3 \) cm
- Let \( BC = 4 \) cm
- Let \( AC = 8 \) cm

2. **Apply the triangle inequality theorem**:
- We need to check the following inequalities:
1. \( AB + BC > AC \)
2. \( AB + AC > BC \)
3. \( BC + AC > AB \)

3. **Check the first inequality**:
- \( AB + BC = 3 + 4 = 7 \)
- Compare with \( AC \):
- \( 7 > 8 \) is **false**.

4. **Check the second inequality**:
- \( AB + AC = 3 + 8 = 11 \)
- Compare with \( BC \):
- \( 11 > 4 \) is **true**.

5. **Check the third inequality**:
- \( BC + AC = 4 + 8 = 12 \)
- Compare with \( AB \):
- \( 12 > 3 \) is **true**.

6. **Conclusion**:
- Since the first inequality \( AB + BC > AC \) is false, we cannot construct a triangle with the given side lengths \( AB = 3 \) cm, \( BC = 4 \) cm, and \( AC = 8 \) cm.

### Final Answer:
No, we cannot construct triangle ABC with the given side lengths.