Power of a point: tangent–secant relation\nIn a circle, chord AB is extended to meet the tangent DE at an external point D. If AB = 12 cm and the tangent length DE = 8 cm, find the length of AD (in cm).

Introduction / Context:
This uses the power of a point theorem: from an external point, the square of the tangent length equals the product of the external and total lengths of a secant through that point. Adjusting the query to AD (the external segment) aligns with standard statements and the given options.

Given Data / Assumptions:

• Chord AB = 12 cm; extended line meets tangent DE at D.
• Tangent length from D: DE = 8 cm.
• Let AD be the external segment; DB = AD + AB.

Concept / Approach:
Power of a point: (tangent)^2 = (external secant) * (entire secant). Hence, DE^2 = AD * (AD + AB). Solve for AD using a quadratic relation.

Step-by-Step Solution:

Verification / Alternative check:
If AD = 4, then DB = 4 + 12 = 16 and 4 * 16 = 64 = DE^2, confirming the theorem.

Why Other Options Are Wrong:
5, 6, and 7 fail the equation AD(AD + 12) = 64. Only 4 satisfies it.

Common Pitfalls:
Assuming chord length enters directly without forming the total secant (external + chord) or misapplying the tangent–secant relation leads to incorrect values.

Final Answer:
4 cm