Circle: A collection of all points in a plane at a constant distance (radius) from a fixed point (centre).
Non-intersecting line: A line that has no common points with the circle.
Secant: A line that intersects a circle at exactly two points.
Tangent: A line that intersects a circle at only one point.
10.2 Tangent to a Circle
Properties of Tangents
A tangent is a limiting case of a secant when the two endpoints of its corresponding chord coincide.
There can be at most two tangents parallel to a given secant.
Point of Contact: The common point where the tangent touches the circle.
Theorem 10.1
Theorem 10.1: The tangent at any point of a circle is perpendicular to the radius through the point of contact.
Normal: The line containing the radius through the point of contact.
10.3 Number of Tangents from a Point on a Circle
Position-Based Tangents
Case 1 (Inside): Zero tangents can be drawn from a point lying inside a circle.
Case 2 (On): Exactly one tangent can be drawn from a point lying on the circle.
Case 3 (Outside): Exactly two tangents can be drawn from a point lying outside the circle.
Length of Tangent: The length of the segment of the tangent from the external point to the point of contact.
Theorem 10.2
Theorem 10.2: The lengths of tangents drawn from an external point to a circle are equal.
Proof Summary: For external point P and tangent points Q and R, triangles ΔOQP and ΔORP are congruent by RHS rule (OQ = OR as radii, OP is common, and ∠OQP = ∠ORP = 90°). Thus, PQ = PR.
10.4 Summary of Key Concepts
Core Takeaways
The tangent is perpendicular to the radius at the point of contact.
Tangents from an external point to a circle are equal in length.
The centre of the circle lies on the angle bisector of the angle between two external tangents.
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