In a circle, chord AB is produced to meet a tangent at point P outside the circle. From the same point P, PT is drawn as a tangent touching the circle at T. If PT = 6 cm and PB = 4 cm, find the length of PA using the tangent–secant theorem.

Introduction / Context:
This problem is about the power of a point theorem in circle geometry. When a tangent and a secant are drawn from an external point to a circle, there is a specific relationship between the tangent length and the product of the external and total secant lengths. Using this relationship, we can find unknown line segments connected to the circle.

Given Data / Assumptions:

• PT is a tangent from external point P, touching the circle at T
• Chord AB is extended to meet the tangent at P, forming a secant PAB
• Length PT = 6 cm
• Length PB = 4 cm (external segment of the secant)
• We must find PA, the full length of the secant from P to A

Concept / Approach:
The tangent–secant theorem states: (length of tangent)^2 = (external part of secant) * (entire secant). In symbols, PT^2 = PB * PA. All lengths are measured from the same external point P. Using the given values, we can solve for the unknown PA by substituting and rearranging the equation.

Step-by-Step Solution:
Apply tangent–secant theorem: PT^2 = PB * PA Substitute PT = 6 cm, PB = 4 cm: 6^2 = 4 * PA 36 = 4 * PA PA = 36 / 4 = 9 cm

Verification / Alternative check:
Substitute PA = 9 back into the relation: PB * PA = 4 * 9 = 36. The square of the tangent length is 6^2 = 36. Since both sides match, the computed value PA = 9 cm is consistent with the tangent–secant theorem.

Why Other Options Are Wrong:
12 cm, 11 cm, 10 cm, and 8 cm: Substituting any of these into PB * PA would not give 36. For example, 4 * 10 = 40 and 4 * 8 = 32, which do not equal PT^2.

Common Pitfalls:
Confusing PB with PA or mixing up which segment is the external part. Thinking PT * PB = PA instead of PT^2 = PB * PA. Forgetting that the entire secant length PA includes both the external part PB and the internal segment BA.

Final Answer:
The length of PA is 9 cm