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

Using the Intersecting Chords Theorem
According to the intersecting chords theorem, when two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

Thus,
\[ AP \times PB = CP \times PD \]
Substituting the given values
\[ 4 \times 6 = 3 \times PD \] \[ 24 = 3PD \]
Solving for \(PD\)
\[ PD = \frac{24}{3} \] \[ PD = 8 \]
Final Answer
The length of \(PD\) is 8 cm.