Number of tangents from a point to a circle | Theorem 10.2 and its applications. | CIRCLES EX 10.2

Number of Tangents from a Point on a Circle To get an idea of the number of tangents from a point on a circle, let us perform the following Theorem 10.2 : The lengths of tangents drawn from an external point to a circle are equal. Proof : We are given a circle with centre O, a point P lying outside the circle and two tangents PQ, PR on the circle from P (see Fig. 10.7). We are required to prove that PQ = PR. For this, we join OP, OQ and OR. Then ∠ OQP and ∠ ORP are right angles, because these are angles between the radii and tangents, and according to Theorem 10.1 they are right angles. Now in right triangles OQP and ORP, OQ = OR (Radii of the same circle) OP = OP (Common) Therefore, ∆ OQP ≅ ∆ ORP (RHS) This gives PQ = PR