AB and CD are two chords of a circle intersecting at a point P inside the circle. If
AB= 24cm, AP= 4cm and PD = 8cm, determine CP

Answer

To solve the problem, we will use the property of intersecting chords in a circle. The property states that if two chords AB and CD intersect at a point P inside the circle, then:

\[ \frac{AP}{CP} = \frac{PD}{PB} \]

Given:
- \( AB = 24 \, \text{cm} \)
- \( AP = 4 \, \text{cm} \)
- \( PD = 8 \, \text{cm} \)

We need to find \( CP \).

### Step-by-Step Solution:

1. **Identify the lengths**:
- We know \( AP = 4 \, \text{cm} \) and \( PD = 8 \, \text{cm} \).
- Let \( CP = x \, \text{cm} \).
- To find \( PB \), we can use the total length of chord AB:
\[ PB = AB - AP = 24 \, \text{cm} - 4 \, \text{cm} = 20 \, \text{cm} \]

2. **Set up the proportion**:
- Using the property of intersecting chords:
\[ \frac{AP}{CP} = \frac{PD}{PB} \]
- Substitute the known values:
\[ \frac{4}{x} = \frac{8}{20} \]

3. **Cross-multiply to solve for \( x \)**:
- Cross-multiplying gives:
\[ 4 \cdot 20 = 8 \cdot x \]
- This simplifies to:
\[ 80 = 8x \]

4. **Solve for \( x \)**:
- Divide both sides by 8:
\[ x = \frac{80}{8} = 10 \, \text{cm} \]

Thus, the length of \( CP \) is \( 10 \, \text{cm} \).

### Final Answer:
\[ CP = 10 \, \text{cm} \]