Find the value of x in the given figure.

Answer

To find the value of \( x \) in the given figure, we can use the Power of a Point theorem, which states that for a point \( P \) outside a circle, the product of the lengths of the segments of any secant line through \( P \) is equal to the product of the lengths of the segments of any other secant line through \( P \).

### Step-by-Step Solution:

1. **Identify the segments**:
- From the figure, we have the following segments:
- \( PC = 8 \)
- \( CA = 4 \)
- \( PD = 7 \)
- \( BD = x \)

2. **Calculate \( PA \)**:
- Since \( PA = PC + CA \), we can calculate:
\[ PA = 8 + 4 = 12 \]

3. **Set up the equation using the Power of a Point theorem**:
- According to the theorem:
\[ PC \times PA = PD \times PB \]
- Here, \( PB = PD + BD = 7 + x \).
- Plugging in the values, we get:
\[ 8 \times 12 = 7 \times (7 + x) \]

4. **Calculate the left side**:
- Calculate \( 8 \times 12 \):
\[ 8 \times 12 = 96 \]

5. **Set up the equation**:
- Now we have:
\[ 96 = 7 \times (7 + x) \]

6. **Distribute on the right side**:
- Distributing \( 7 \):
\[ 96 = 49 + 7x \]

7. **Isolate \( 7x \)**:
- Subtract \( 49 \) from both sides:
\[ 96 - 49 = 7x \]
\[ 47 = 7x \]

8. **Solve for \( x \)**:
- Divide both sides by \( 7 \):
\[ x = \frac{47}{7} \approx 6.71 \]

9. **Final answer**:
- Thus, the value of \( x \) is approximately \( 6.71 \).