⭕ Tangent-Secant-Chord

Circle power problems are common in AMC geometry. Master these patterns for efficient circle problem solving.

🎯 Recognition Cues

Key Triggers

  • Tangents - Look for lines touching circles at exactly one point
  • Secants - Look for lines intersecting circles at two points
  • Chords - Look for line segments connecting two points on circles
  • Power of a Point - Look for products of lengths from external points

Common Setups

  • External point with tangents and secants
  • Intersecting chords or secants
  • Tangent-secant configurations
  • Equal tangent lengths

🧩 Solution Template

Step 1: Identify the Configuration

  • Determine if you have tangents, secants, or chords
  • Identify the external point (if any)
  • Mark given lengths and relationships

Step 2: Apply Power of a Point

  • Tangent-Secant: $PA^2 = PB \cdot PC$
  • Two Secants: $PA \cdot PB = PC \cdot PD$
  • Two Chords: $PA \cdot PB = PC \cdot PD$

Step 3: Set Up the Equation

  • Write the Power of a Point equation
  • Substitute known values
  • Solve for the unknown

Step 4: Verify

  • Check that the answer makes geometric sense
  • Ensure all lengths are positive
  • Verify the Power of a Point relationship

🔍 Worked Example

Problem: In the figure, $PA$ is tangent to the circle at $A$, $PB$ is a secant intersecting the circle at $B$ and $C$, and $PA = 6$, $PB = 4$. Find $BC$.

Solution: Step 1: Identify configuration

  • $PA$ is tangent to circle at $A$
  • $PB$ is secant intersecting circle at $B$ and $C$
  • $PA = 6$, $PB = 4$

Step 2: Apply Power of a Point

  • This is a tangent-secant configuration
  • Use formula: $PA^2 = PB \cdot PC$

Step 3: Set up equation

  • $PA^2 = PB \cdot PC$
  • $6^2 = 4 \cdot PC$
  • $36 = 4 \cdot PC$
  • $PC = 9$

Step 4: Find $BC$

  • Since $PC = PB + BC$ and $PB = 4$
  • $9 = 4 + BC$
  • $BC = 5$

Answer: $BC = 5$

⚠️ Common Pitfalls

Pitfall: Wrong Power of a Point formula

  • Fix: Remember that tangent-secant uses $PA^2 = PB \cdot PC$

Pitfall: Confusing tangent and secant

  • Fix: Tangent touches circle at one point, secant intersects at two points

Pitfall: Wrong segment identification

  • Fix: Make sure you’re using the correct segments in the formula

Pitfall: Forgetting about equal tangents

  • Fix: From external point, two tangents are always equal in length

🔗 Related Patterns

💡 Quick Reference

Power of a Point Formulas

  • Tangent-Secant: $PA^2 = PB \cdot PC$
  • Two Secants: $PA \cdot PB = PC \cdot PD$
  • Two Chords: $PA \cdot PB = PC \cdot PD$

Common Properties

  • Equal Tangents: From external point, two tangents are equal
  • Chord Length: $2\sqrt{r^2 - d^2}$ where $r$ is radius, $d$ is distance from center
  • Tangent Properties: Tangent is perpendicular to radius at point of contact

Solution Strategy

  • Identify: Determine tangent, secant, or chord configuration
  • Apply: Use appropriate Power of a Point formula
  • Calculate: Solve for unknown lengths
  • Verify: Check that answer makes geometric sense

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