Circles

Chords, Tangents & Secants

Finding lengths formed by chords, tangent lines, and secant lines of a circle.

Advanced20 min lesson3 min readUpdated August 12, 2026Author not yet attributed

Prerequisites

Central & Inscribed Angles

Lines That Touch or Cross a Circle

A tangent line touches a circle at exactly one point. Before reading on, think about the radius drawn to that exact touching point — predict what angle it must form with the tangent line, and why a line could only touch a circle once if that were true.

Definition — Chord, Tangent, and Secant

A chord is a segment connecting two points on a circle. A tangent line touches a circle at exactly one point. A secant line crosses a circle at two points.

A tangent line is always perpendicular to the radius drawn to its point of tangency — and here's why it must be: the radius is the shortest possible segment from the center to any point on the tangent line, since every other point on that line lies genuinely outside the circle (farther from the center than the radius). The shortest segment from a point to a line is always the perpendicular one, so the radius-to-tangent-point segment must be that perpendicular segment.

Several length relationships follow from circles' symmetry. Two tangent segments drawn to a circle from the same external point are always congruent — a consequence of two right triangles sharing a hypotenuse (from the external point to the center) and a leg (the radius), forcing their remaining legs to match. When two chords intersect inside a circle, the products of their two segment pieces are always equal.

Worked Example — Using the Two-Tangent Congruence Property

Two tangent segments are drawn to a circle from the same external point P, touching the circle at points A and B. If PA = 12, find PB. Since two tangent segments from the same external point are always congruent: PB = 12.

Worked Example — Using the Intersecting Chords Relationship

Two chords intersect inside a circle. One chord is split into pieces of length 4 and 9; the other is split into pieces of length 6 and x. Find x. The products of each chord's pieces must be equal: 4 × 9 = 6 × x, so 36 = 6x, giving x = 6.

Geometry Canvas

Point mode (click to add a point)LabelsMeasurementsSnap to grid

Construct

Tool

Objects

1.
Color
P1
P1: a free point, draggable on the plane
2.
Color
P2
P2: a free point, draggable on the plane
3.
Color
P3
P3: a free point, draggable on the plane
4.
Color
poly1
poly1: the polygon through P1, P2, P3

Measurements

poly1area = 15perimeter = 17.66

Tip

Whenever a tangent line and a radius meet at the point of tangency, mark that angle as a right angle immediately — it's one of the most reliably useful facts for setting up a Pythagorean Theorem relationship in a circle problem.

Common Mistakes

Assuming any two tangent segments to a circle are congruent, without checking that they're drawn from the same external point.
The two-tangent congruence property specifically requires both tangent segments to originate from the same external point — tangent segments from two different points have no guaranteed relationship.
Adding the two chord pieces instead of multiplying them when applying the intersecting chords relationship.
The intersecting chords relationship equates the products of each chord's two pieces, not their sums — 4 × 9 must equal 6 × x, not 4 + 9 equaling 6 + x.

Key Takeaways

A tangent line is always perpendicular to the radius drawn to its point of tangency.
Two tangent segments from the same external point are always congruent.
When two chords intersect inside a circle, the products of their segment pieces are equal.

Summary

Chords, tangents, and secants reveal predictable length relationships rooted in a circle's symmetry. The next unit shifts from circle theorems to measuring the area and volume of two- and three-dimensional figures.

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