Problem Statement
In the given figure, $PT$ is a tangent to the circle drawn from an external point $P$, touching the circle at $T$. A straight secant line through $P$ intersects the circle at points $A$ and $B$ (with $A$ lying between $P$ and $B$). If $PT = 8\text{ cm}$ and $PA = 4\text{ cm}$, calculate:
(i) The length of the entire secant segment $PB$.
(ii) The length of the chord $AB$.
Verified Solution & Marking Scheme
Apply the Tangent-Secant Theorem
By the geometric Tangent-Secant Power Theorem:
$$PT^2 = PA \cdot PB$$
Substitute the given values $PT = 8\text{ cm}$ and $PA = 4\text{ cm}$:
$$8^2 = 4 \cdot PB \implies 64 = 4 \cdot PB$$
$$PB = \frac{64}{4} = 16\text{ cm}$$
Find length of chord AB
Since $A$ lies on the line segment $PB$:
$$PB = PA + AB \implies 16 = 4 + AB$$
$$AB = 16 - 4 = 12\text{ cm}$$