Segment Lengths: Intersecting Chords
When two chords cross inside a circle, the products of their segments are equal: .
Introduction
Products of chord segments are equal
Past Knowledge
Chord angles (10.2.2). Similar triangles (6.1). Cross-multiplication.
Today's Goal
Prove and apply the intersecting chords segment theorem.
Future Success
Secant/tangent segments (10.2.5), power of a point.
Key Concepts
Intersecting Chords Segment Theorem
If two chords intersect at point P inside a circle:
“The products of the segments of each chord are equal.”
Theorem & Proof
Proof: Intersecting Chords → Equal Products
Given: Chords and intersect at
Prove:
| # | Statement | Reason |
|---|---|---|
| 1 | Vertical angles | |
| 2 | Inscribed angles intercepting same arc | |
| 3 | AA Similarity (steps 1, 2) | |
| 4 | → | Corresponding sides proportional → cross-multiply |
∎ Similar triangles from inscribed angles give proportional sides, and cross-multiplying yields the product rule.
Worked Examples
Finding a Segment
Chords intersect: segments 6, 8, 4, x. Find x.
→
x = 12
Segment in Terms of x
Chord AB: AP = x, PB = 9. Chord CD: CP = x + 1, PD = 6. Find x.
→ →
Segments: AP = 2, PB = 9, CP = 3, PD = 6. Check: 2 × 9 = 18 = 3 × 6 ✓
x = 2
Quadratic
Chord segments: x, (x + 2), 3, 8. Find x.
→
(reject x = −6, length can't be negative)
x = 4
Common Pitfalls
Multiplying Wrong Pairs
Multiply the two segments of the SAME chord, not segments from different chords on the same side.
Rejecting Negative Solutions
When solving quadratics, remember that lengths must be positive. Always reject negative solutions.
Real-Life Applications
Tunnel & Pipe Engineering
When a tunnel (circular cross-section) intersects a geological fault line at two points, the chord-product theorem helps calculate the fault's depth below the tunnel crown.
Archaeology — Pottery Reconstruction
Archaeologists use the chord product rule to estimate the original diameter of a broken circular pot from just a fragment and two intersecting measurements.
Practice Quiz
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