Equal Tangent Lengths Theorem

Theorem 10.2: Proving Equal Tangent Lengths from External Points

Theorem 10.2
The lengths of tangents drawn from an external point to a circle are equal.
Current Step: Setup
Given: Circle O, Point P
Tangent PQ: --
Tangent PR: --
Angle ∠QPR: --
Proof Status: In Progress
Welcome to the Equal Tangent Lengths Theorem! We'll explore two different methods to prove this fundamental relationship.
Method 1: Congruent Triangles
  • ∠OQP = ∠ORP = 90° (tangent ⊥ radius)
  • OQ = OR (equal radii)
  • OP = OP (common side)
  • △OQP ≅ △ORP (RHS congruence)
  • Therefore: PQ = PR (CPCT)
Method 2: Pythagoras Theorem
  • PQ² = OP² - OQ² (right triangle OQP)
  • PR² = OP² - OR² (right triangle ORP)
  • Since OQ = OR (equal radii)
  • Therefore: PQ² = PR²
  • Hence: PQ = PR