In a triangle ABC, if AB,BC and AC are the three sides of the triangle, then which of the statements is necessarily?

Options

A. `AB+BCltAC`
B. `AB+BCgtAC`
C. AB+BC=AC
D. `AB^2+BC^2=AC^2`

Answer

To solve the problem regarding the triangle ABC with sides AB, BC, and AC, we need to determine which statements about the sides are necessarily true.

### Step-by-Step Solution:

1. **Understanding Triangle Inequality Theorem**:
The triangle inequality 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. This can be expressed in three inequalities:
- \( AB + BC > AC \)
- \( AB + AC > BC \)
- \( BC + AC > AB \)

2. **Analyzing the Statements**:
We are given four statements regarding the sides of the triangle. We need to identify which of these statements is necessarily true based on the triangle inequality theorem.

3. **Applying the Triangle Inequality**:
- For the first statement, check if \( AB + BC > AC \) holds true. This is true for any triangle.
- For the second statement, check if \( AB + AC > BC \) holds true. This is also true for any triangle.
- For the third statement, check if \( BC + AC > AB \) holds true. This is again true for any triangle.
- The fourth statement may refer to a specific condition (like being a right triangle), which is not necessarily true for all triangles.

4. **Conclusion**:
Based on the triangle inequality theorem, the first three statements are necessarily true for any triangle ABC. The fourth statement may not hold for all triangles, hence it is not necessarily true.

### Final Answer:
The statements that are necessarily true for triangle ABC are:
- \( AB + BC > AC \)
- \( AB + AC > BC \)
- \( BC + AC > AB \)