In a `Delta ABC, angle C = 110^(@)`. Which one of the following statements is correct ?

Options

A. `AB^(2) gt AC^(2) + BC^(2)`
B. `AB^(2) lt AC^(2) + BC^(2)`
C. `AC^(2) gt AB^(2) + BC^(2)`
D. `BC^(2) gt AB^(2) + AC^(2)`

Answer

To solve the problem regarding triangle ABC where angle C = 110°, we will analyze the statements related to the triangle using the Cosine Rule.

### Step-by-Step Solution:

1. **Understanding the Triangle and Given Information:**
We have triangle ABC with angle C = 110°. We need to determine which statement about this triangle is correct.

**Hint:** Remember that the sum of angles in a triangle is always 180°.

2. **Using the Cosine Rule:**
The Cosine Rule states that in any triangle ABC:
\[ c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \]
Here, \(c\) is the side opposite angle C, \(a\) is the side opposite angle A, and \(b\) is the side opposite angle B. For our triangle, we can denote:
- \(AB = c\)
- \(AC = b\)
- \(BC = a\)

3. **Applying the Cosine Rule:**
Plugging in the values we have:
\[ AB^2 = AC^2 + BC^2 - 2 \cdot AC \cdot BC \cdot \cos(110°) \]

4. **Calculating Cosine of 110°:**
Since \(110°\) is in the second quadrant, we know that:
\[ \cos(110°) = -\cos(70°) \]
Therefore, we can rewrite the equation as:
\[ AB^2 = AC^2 + BC^2 + 2 \cdot AC \cdot BC \cdot \cos(70°) \]

5. **Conclusion about the Sides:**
Since \( \cos(70°) \) is positive, the term \( 2 \cdot AC \cdot BC \cdot \cos(70°) \) will also be positive. This means:
\[ AB^2 > AC^2 + BC^2 \]
Therefore, the side opposite the largest angle (which is angle C = 110°) is longer than the sum of the squares of the other two sides.

6. **Final Statement:**
Based on the analysis, the correct statement is that in triangle ABC, \(AB\) (the side opposite angle C) is greater than the sum of the squares of the other two sides.