AB and CD are two chords of a circle intersecting at a point P outside the circle. If
PC=30cm, CD=14cm and PA=24cm, determine AB.

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 outside the circle, then the following relationship holds:

\[ PA \times PB = PC \times PD \]

Where:
- PA = distance from point P to point A
- PB = distance from point P to point B
- PC = distance from point P to point C
- PD = distance from point P to point D

Given:
- \( PC = 30 \, \text{cm} \)
- \( CD = 14 \, \text{cm} \)
- \( PA = 24 \, \text{cm} \)

We need to find the length of chord AB.

### Step-by-step Solution:

1. **Identify the lengths**:
- We know \( PA = 24 \, \text{cm} \) and \( PC = 30 \, \text{cm} \).
- The length of chord CD is given as \( CD = 14 \, \text{cm} \).

2. **Calculate PD**:
- Since \( CD = 14 \, \text{cm} \), we can find \( PD \) using the relationship:
\[ PD = CD - PC = 30 - 14 = 16 \, \text{cm} \]

3. **Set up the equation**:
- Let \( PB = x \). Then, we can express \( PB \) in terms of \( AB \):
\[ PB = AP - AB = PA - AB = 24 - AB \]

4. **Apply the intersecting chords property**:
- According to the property:
\[ PA \times PB = PC \times PD \]
Substituting the known values:
\[ 24 \times (24 - AB) = 30 \times 16 \]

5. **Calculate the right-hand side**:
- Calculate \( 30 \times 16 \):
\[ 30 \times 16 = 480 \]

6. **Expand and simplify the equation**:
\[ 24 \times (24 - AB) = 480 \]
Expanding the left side:
\[ 576 - 24AB = 480 \]

7. **Rearranging the equation**:
\[ 576 - 480 = 24AB \]
\[ 96 = 24AB \]

8. **Solve for AB**:
\[ AB = \frac{96}{24} = 4 \, \text{cm} \]

Thus, the length of chord AB is \( 4 \, \text{cm} \).