ABC is a triangle right- angled at B and D is a point on BC produced `(BD gt BC)`, such that `BD=2DC.` Which one of the following is correct?

Options

A. `AC^(2)=AD^(2)-3CD^(2)`
B. `AC^(2)= AD^(2)-2CD^(2)`
C. `AC^(2)=AD^(2)-4CD^(2)`
D. `AC^(2)=AD^(2)-5CD^(2)`

Answer

To solve the problem, we need to analyze the triangle ABC, which is right-angled at B, and the point D on the extended line BC such that \( BD > BC \) and \( BD = 2DC \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
- We know that triangle ABC is right-angled at B.
- We have a point D such that \( BD = 2DC \).
- Let \( DC = x \). Then, \( BD = 2x \) and \( BC = BD - DC = 2x - x = x \).

2. **Setting Up the Equation:**
- From the above, we can conclude that \( BC = x \) and \( DC = x \).
- Thus, \( BD = 2DC = 2x \).

3. **Using the Pythagorean Theorem:**
- For triangle ABC (right-angled at B):
\[ AC^2 = AB^2 + BC^2 \quad \text{(1)} \]
- For triangle ABD (right-angled at B):
\[ AD^2 = AB^2 + BD^2 \quad \text{(2)} \]

4. **Substituting Values:**
- From equation (1):
\[ AC^2 = AB^2 + x^2 \]
- From equation (2):
\[ AD^2 = AB^2 + (2x)^2 = AB^2 + 4x^2 \]

5. **Finding the Relation Between AC and AD:**
- Now, we can set up the relationship:
\[ AC^2 - AD^2 = (AB^2 + x^2) - (AB^2 + 4x^2) \]
- This simplifies to:
\[ AC^2 - AD^2 = x^2 - 4x^2 = -3x^2 \]

6. **Finalizing the Expression:**
- Rearranging gives:
\[ AC^2 = AD^2 - 3x^2 \]

### Conclusion:
The correct relationship derived from the problem is:
\[ AC^2 = AD^2 - 3 \cdot DC^2 \]