Number of Tangents from a Point
Point Inside Circle
0 tangents can be drawn (any line through internal point is a secant).Point On Circle
Exactly 1 tangent can be drawn touching the circle at that point.Point Outside Circle
Exactly 2 tangents can be drawn from the external point to the circle.Theorem 10.2 & Complete Proof
Tangents PQ and PR drawn from external point P to circle C(O, r)
To Prove: PQ = PR
Join OP, OQ, and OR. By Theorem 10.1, ∠OQP = ∠ORP = 90°.
In right-angled triangles △OQP and △ORP:
1. OQ = OR (Radii of same circle)
2. OP = OP (Common Hypotenuse)
By RHS Congruence Criterion, △OQP ≅ △ORP.
By CPCT (Corresponding Parts of Congruent Triangles), PQ = PR (Proved).
Key Properties of External Tangents
∠POQ = ∠POR (The two tangents subtend equal angles at the center of the circle).
∠QPO = ∠RPO (The two tangents are equally inclined to the line segment joining the center to the point P).
Step-by-Step Problem Solving
If tangents PA and PB from a point P to a circle with center O are inclined to each other at angle of 80°, find ∠POA
Given: Total angle ∠APB = 80°. By symmetry (∠APO = ∠BPO):
In right △OAP (∠OAP = 90°):
∠POA + ∠APO + ∠OAP = 180°
∠POA + 40° + 90° = 180° ⟹ ∠POA = 50°
Key Takeaways
From an external point, exactly two tangents of equal length can be drawn (PQ = PR).
Proof uses RHS Congruence of right triangles △OQP and △ORP.
Tangents subtend equal angles at center (∠POQ = ∠POR) and are equally inclined to OP.