Problem Statement
In the given circle, chords $AB$ and $CD$ intersect internally at $P$. If $AP = 8\text{ cm}, PB = 3\text{ cm}$ and $CP = 4\text{ cm}$, find $PD$. If $PT$ is a tangent drawn from an external point $P'$ on secant $P'AB$ such that $P'A = 4\text{ cm}$ and $AB = 5\text{ cm}$, find the length of tangent $P'T$.
Verified Solution & Marking Scheme
Intersecting Chords Theorem for PD
When two chords intersect internally at $P$:
$AP \times PB = CP \times PD$
$8 \times 3 = 4 \times PD \implies 24 = 4 PD \implies PD = 6\text{ cm}$
Tangent-Secant Theorem for Tangent P'T
For external point $P'$ on secant $P'AB$:
$P'T^2 = P'A \times P'B$
Here $P'A = 4\text{ cm}$ and $P'B = P'A + AB = 4 + 5 = 9\text{ cm}$.
Calculate Length of Tangent P'T
$P'T^2 = 4 \times 9 = 36 \implies P'T = \sqrt{36} = 6\text{ cm}$