Power of a point: tangent–secant relation In 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).
-
A4 cm
-
B5 cm
-
C6 cm
-
D7 cm
-
ENone of these
Answer
Correct Answer: 4 cm
Explanation
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:
DE^2 = AD * (AD + AB) ⇒ 8^2 = AD * (AD + 12) 64 = AD^2 + 12AD ⇒ AD^2 + 12AD − 64 = 0 Discriminant = 12^2 + 4 * 64 = 400 ⇒ AD = (−12 + 20)/2 = 4 cm (positive root)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