Click image to open full size
Segment Relationships in Circles
Chords, Secants, and Tangents
Related Tools
Segment relationships in circles connect geometry diagrams to exact equations for missing lengths. These theorems help students solve problems involving chords, secants, and tangents without measuring directly. They are important because many circle problems look different on the surface but follow the same small set of patterns. Learning to identify the pattern is the key step in solving them correctly.
Each theorem compares products or equal lengths created by lines that intersect a circle. For intersecting chords, the products of the chord segments are equal. For secants and tangents drawn from the same outside point, the outside part and the whole length work together in a predictable equation. Once you label the segments carefully, you can translate the picture into an algebra equation and solve for the unknown.
Key Facts
- Intersecting chords theorem: if two chords intersect inside a circle, then a*b = c*d.
- Secant-secant theorem: if two secants are drawn from the same outside point, then external1(whole1) = external2(whole2).
- Secant-tangent theorem: if a secant and a tangent are drawn from the same outside point, then external(whole) = tangent^2.
- Tangent-tangent theorem: if two tangents are drawn from the same outside point, then t1 = t2.
- Whole secant length = external segment + internal segment.
- A tangent touches a circle at exactly one point, while a secant cuts through the circle at two points.
Vocabulary
- Chord
- A chord is a line segment whose endpoints both lie on the circle.
- Secant
- A secant is a line that intersects a circle at two points.
- Tangent
- A tangent is a line that touches a circle at exactly one point.
- External segment
- An external segment is the part of a secant outside the circle from the outside point to the first intersection.
- Whole secant
- A whole secant is the entire distance from the outside point through the circle to the far intersection point.
Common Mistakes to Avoid
- Using only the inside part of a secant as the whole length, which is wrong because the theorem uses the entire secant from the outside point to the far intersection.
- Mixing up the intersecting chords theorem with the secant-secant theorem, which is wrong because chords intersect inside the circle while secants start from a point outside the circle.
- Adding segment lengths when the theorem requires multiplication, which is wrong because these circle relationships are product equations such as a*b = c*d.
- Assuming any two segments from an outside point are equal, which is wrong because only two tangents from the same outside point have equal lengths.
Practice Questions
- 1 Two chords intersect inside a circle. One chord is split into segments of lengths 4 and 9. The other chord is split into segments of lengths 6 and x. Find x.
- 2 From a point outside a circle, one secant has an external segment of 5 and an internal segment of 7. A tangent from the same point has length t. Find t.
- 3 A student writes the equation 3*8 = 4*x for a diagram with two secants drawn from the same outside point, where 8 and x are only the inside parts of the secants. Explain why this setup is incorrect and describe what lengths should be used instead.