Segment Lengths in Circles
Overview
When chords, secants, and tangents meet, their segment lengths are locked into simple products. Inside the circle, the products of the segments of two intersecting chords are equal. Outside the circle, each secant multiplies as (external segment) × (whole secant), and a tangent segment squared equals that same product. Every problem in this lesson is setting two of those products equal.
Vocabulary
- Secant Segment
- A segment with one endpoint at an exterior point and the other endpoint on the circle, lying along a secant line.
- External Segment
- The part of a secant segment that lies outside the circle, from the exterior point to the nearer intersection point with the circle.
- Whole Secant
- The full secant segment from the exterior point to the farther intersection point with the circle: external segment + internal segment.
- Tangent Segment
- A segment with one endpoint at an exterior point and the other at the point of tangency on the circle.
Main Ideas
- Segments of Chords: if two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.
- Segments of Secants: if two secant segments share an endpoint outside a circle, then (external)(whole) for one secant equals (external)(whole) for the other.
- Segments of Secants and Tangents: if a secant segment and a tangent segment share an exterior endpoint, then (tangent)² = (external)(whole secant).
- The most common error is using the internal piece where the whole secant belongs. Build the whole secant first — external + internal — before you multiply.
Lesson Notes
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Lesson Video
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Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.