AB and CD are two chords of a circle intersecting at a point P outside the circle. If
PA= 8cm, PC= 5cm and PD= 4cm, determine AB.

Answer

To solve the problem, we will use the intersecting chords theorem, which states that if two chords intersect, the products of the lengths of the segments of each chord are equal.

### Step-by-Step Solution:

1. **Identify the segments of the chords:**
- Let \( PA = 8 \, \text{cm} \)
- Let \( PC = 5 \, \text{cm} \)
- Let \( PD = 4 \, \text{cm} \)
- Let \( AB = x \, \text{cm} \)
- Therefore, \( PB = PA - AB = 8 - x \, \text{cm} \)

2. **Apply the intersecting chords theorem:**
According to the theorem, we have:
\[ PA \times PB = PC \times PD \]
Substituting the known values:
\[ 8 \times (8 - x) = 5 \times 4 \]

3. **Calculate the right side:**
\[ 5 \times 4 = 20 \]
So the equation becomes:
\[ 8 \times (8 - x) = 20 \]

4. **Expand the left side:**
\[ 64 - 8x = 20 \]

5. **Rearrange the equation:**
Subtract 64 from both sides:
\[ -8x = 20 - 64 \]
\[ -8x = -44 \]

6. **Solve for \( x \):**
Divide both sides by -8:
\[ x = \frac{-44}{-8} = 5.5 \, \text{cm} \]

7. **Conclusion:**
The length of chord \( AB \) is \( 5.5 \, \text{cm} \).