Intersecting Chords — Definition, Formula & Examples
Intersecting chords are two chords of a circle that cross each other at a point inside the circle. The intersecting chords theorem states that the products of the two segments of each chord are always equal.
If two chords and of a circle intersect at an interior point , then . This relationship holds for any pair of chords that meet inside the circle, regardless of their lengths or angles.
Key Formula
Where:
- = The point where the two chords intersect inside the circle
- = Distance from endpoint A to the intersection point
- = Distance from the intersection point to endpoint C
- = Distance from endpoint B to the intersection point
- = Distance from the intersection point to endpoint D
How It Works
When two chords cross inside a circle, the intersection point splits each chord into two segments. Multiply the two segment lengths of one chord together, then do the same for the other chord — the two products are always equal. This works because the two triangles formed by connecting the chord endpoints are similar, which forces the segment lengths into a specific proportion.
Worked Example
Problem: Two chords intersect inside a circle. One chord is split into segments of length 3 and 8. The other chord is split into segments of length 4 and x. Find x.
Apply the theorem: Set the products of the segments equal to each other.
Solve for x: Simplify the left side and divide both sides by 4.
Answer: The unknown segment length is .
Why It Matters
The intersecting chords theorem appears frequently on standardized tests and in high school geometry courses. It is also foundational for the power of a point concept, which unifies several circle theorems used in competition mathematics and analytic geometry.
Common Mistakes
Mistake: Multiplying the two segments of different chords instead of same-chord segments.
Correction: Each product must use the two pieces of the same chord. If chord AC is split into AP and PC, multiply those together — not AP with BP.