Chapter 10: Circles
Exploring Tangents, Secants, and the Magical Properties of Touch.
📖 Lines & Circles: The 3 Possibilities
When a circle and a line are in a plane, only 3 things can happen:
The line and the circle have NO common point.
The line intersects the circle at exactly TWO distinct points.
The line touches the circle at exactly ONE point (Point of Contact).
📐 Theorem 10.1: Perpendicularity
The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Proof Logic: Any point 'Q' on the tangent other than the point of contact 'P' must lie outside the circle. Hence OQ > OP. Since OP is the shortest distance, it MUST be perpendicular!
📏 Theorem 10.2: Equal Tangents
The lengths of tangents drawn from an external point to a circle are exactly equal.
Proof Logic: Draw two tangents PQ and PR. Join OP, OQ, OR. Since ∠OQP = ∠ORP = 90°, ΔOQP ≅ ΔORP by RHS Congruency. Thus PQ = PR (CPCT).
⭐ Board Exam Super-Hacks
The Angle Bisector Trick
The line joining the external point to the centre (OP) bisects the angle between the two tangents. So, ∠OPQ = ∠OPR. It also bisects the central angle: ∠POQ = ∠POR.
Concentric Circles Secret
If a chord of a larger circle touches a smaller concentric circle, it becomes a tangent to the smaller one. Because of Theorem 10.1, the radius bisects this chord at the point of contact!
⚠️ Common Exam Traps
Trap 1: How Many Tangents?
- From INSIDE the circle: 0 tangents.
- From ON the circle: Exactly 1 tangent.
- From OUTSIDE the circle: Exactly 2 tangents.
Trap 2: The "Normal" Word
If a question mentions the "normal" to a circle, don't get confused! It simply means the line containing the radius passing through the point of contact.