Question

In a circle, two chords $AB$ and $CD$ intersect internally at $P$. If $AP = 4$ cm, $PB = 6$ cm, and $CP = 3$ cm, find the length of $PD$.

Hint:

For intersecting chords inside a circle: product of segments of one chord = product of segments of the other.

Answer 1

Concept: When two chords intersect inside a circle, the Intersecting Chords Theorem states: \[ AP \times PB = CP \times PD \]
Step 1: Write the formula. \[ AP \cdot PB = CP \cdot PD \]
Step 2: Substitute the given values. \[ 4 \times 6 = 3 \times PD \]
Step 3: Simplify. \[ 24 = 3 \times PD \]
Step 4: Solve for $PD$. \[ PD = \frac{24}{3} = 8 \text{ cm} \]
Conclusion: The length of $PD$ is 8 cm.