Chords AB and CD of a circle intersect at a point P inside the circle. If AB = 10 cm, AP = 4 cm and PC = 5 cm, then CD is equal to:

Options

A. 4.8 cm
B. 6.8 cm
C. 9.8 cm
D. 7.8 cm

Answer

To solve the problem, we will use the properties of intersecting chords in a circle. The key property we'll use is that the product of the segments of one chord is equal to the product of the segments of the other chord.

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Length of chord AB = 10 cm
- Length of segment AP = 4 cm
- Length of segment PC = 5 cm

2. **Calculate the Length of Segment PB:**
- Since AB = AP + PB, we can express PB as:
\[ PB = AB - AP = 10 \, \text{cm} - 4 \, \text{cm} = 6 \, \text{cm} \]

3. **Set Up the Relationship for Chords:**
- According to the intersecting chords theorem:
\[ AP \times PB = PC \times PD \]
- We know:
- \( AP = 4 \, \text{cm} \)
- \( PB = 6 \, \text{cm} \)
- \( PC = 5 \, \text{cm} \)
- Let \( PD \) be the unknown length we need to find.

4. **Substitute the Known Values into the Equation:**
- Substitute the known values into the equation:
\[ 4 \times 6 = 5 \times PD \]
- This simplifies to:
\[ 24 = 5 \times PD \]

5. **Solve for PD:**
- Rearranging gives:
\[ PD = \frac{24}{5} = 4.8 \, \text{cm} \]

6. **Calculate the Length of CD:**
- Now, we can find the length of chord CD:
\[ CD = PC + PD = 5 \, \text{cm} + 4.8 \, \text{cm} = 9.8 \, \text{cm} \]

7. **Final Answer:**
- Therefore, the length of chord CD is:
\[ \boxed{9.8 \, \text{cm}} \]