Circles — Notes
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10.1 Tangent to a Circle
A tangent is a line that touches a circle at exactly one point. The point is called the point of contact.
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10.2 Radius and Tangent
The radius through the point of contact is perpendicular to the tangent.
Key theorem
If PT is tangent at T to a circle with centre O, then OT ⟂ PT.
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10.3 Tangents from an External Point
From a point outside a circle, exactly two tangents can be drawn to the circle.
Equal tangent lengths
If PA and PB are tangents from an external point P, then PA = PB.
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10.4 Tangent-Based Proofs
Many problems combine right triangles, equal radii and equal tangent lengths. Identify the centre, point of contact and external point before applying a theorem.
⚡ Quick Revision
- Tangent touches at one point.
- Radius to point of contact is perpendicular to tangent.
- Two tangents can be drawn from an external point.
- Tangents from the same external point are equal.
🎯 Board Exam Focus
- State the theorem before using it in a proof.
- Mark right angles at points of contact.
- Use equal radii and equal tangents to establish congruence or lengths.
🚀 Last-Minute Revision
✔ Read every formula with its conditions and symbols.
✔ Write given data first and keep units consistent.
✔ For proofs, state the theorem and show logical steps.
✔ Check signs, substitutions and final units.
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