A line segment AB is 8 cm in length. AB is produced to P such that `BP^(2)=AB.AP,` find the length of BP.

Answer

To solve the problem, we need to find the length of BP given that \( BP^2 = AB \cdot AP \) and \( AB = 8 \) cm.

### Step-by-Step Solution:

1. **Define the Variables**:
Let \( BP = x \) cm. Since \( AB = 8 \) cm, we can express \( AP \) in terms of \( x \):
\[ AP = AB + BP = 8 + x \]

2. **Set Up the Equation**:
According to the problem, we have:
\[ BP^2 = AB \cdot AP \]
Substituting the values we have:
\[ x^2 = 8 \cdot (8 + x) \]

3. **Expand the Equation**:
Expanding the right-hand side:
\[ x^2 = 64 + 8x \]

4. **Rearrange the Equation**:
Rearranging the equation gives us:
\[ x^2 - 8x - 64 = 0 \]

5. **Use the Quadratic Formula**:
The quadratic formula is given by:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Here, \( a = 1 \), \( b = -8 \), and \( c = -64 \).

6. **Calculate the Discriminant**:
First, calculate the discriminant:
\[ b^2 - 4ac = (-8)^2 - 4 \cdot 1 \cdot (-64) = 64 + 256 = 320 \]

7. **Substitute into the Quadratic Formula**:
Now substituting back into the formula:
\[ x = \frac{8 \pm \sqrt{320}}{2} \]

8. **Simplify the Square Root**:
Simplifying \( \sqrt{320} \):
\[ \sqrt{320} = \sqrt{64 \cdot 5} = 8\sqrt{5} \]
So, we have:
\[ x = \frac{8 \pm 8\sqrt{5}}{2} \]
Simplifying further:
\[ x = 4 \pm 4\sqrt{5} \]

9. **Determine the Length of BP**:
Since length cannot be negative, we take the positive value:
\[ BP = 4(1 + \sqrt{5}) \text{ cm} \]

### Final Answer:
The length of \( BP \) is \( 4(1 + \sqrt{5}) \) cm.