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Chapter 10 Class 10 Circles
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Theorem 10.2 - The lengths of tangents drawn from n external point to circle are equal..jpg

Theorem 10.2 - Chapter 10 Class 10 Circles - Part 2

Theorem 10.2 - Chapter 10 Class 10 Circles - Part 3

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Theorem 10.2 - Chapter 10 Class 10 Circles - Part 4

Theorem 10.2 - Chapter 10 Class 10 Circles - Part 5

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Theorem 10.2 (Method 1) The lengths of tangents drawn from an external point to a circle are equal. Given: Let circle be with centre O and P be a point outside circle PQ and PR are two tangents to circle intersecting at point Q and R respectively To prove: Lengths of tangents are equal i.e. PQ = PR Construction: Join OQ , OR and OP Proof: As PQ is a tangent OQ ⊥ PQ So, ∠ OQP = 90° Hence Δ OQP is right triangle Similarly, PR is a tangent & OR ⊥ PR So, ∠ ORP = 90° Hence Δ ORP is right triangle Using Pythagoras theorem (Hypotenuse)2 = (Height)2 + (Base)2 From (1) and (2) PQ2 = PR2 PQ = PR Hence proved Theorem 10.2 (Method 2) The lengths of tangents drawn from an external point to a circle are equal. Given: Let circle be with centre O and P be a point outside circle PQ and PR are two tangents to circle intersecting at point Q and R respectively To prove: Lengths of tangents are equal i.e. PQ = PR Construction: Join OQ , OR and OP Proof: As PQ is a tangent OQ ⊥ PQ So, ∠ OQP = 90° Similarly, PR is a tangent & OR ⊥ PR So, ∠ ORP = 90° In Δ OQP and Δ ORP ∠ OQP = ∠ ORP OP = OP OQ = OR ∴ Δ OQP = Δ ORP Hence, PQ = PR Hence both tangents from external point are equal in length Also, ∠ OPQ = ∠ OPR Hence, OP is the angle bisector of ∠ QPR

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.