Tangents, Chords & Secants

A tangent meets the radius at 90°; intersecting chords multiply equally.

The idea

Three kinds of line can meet a circle. A tangent touches the circle at exactly one point. A secant crosses it at two points, and the segment of a secant between its two crossing points is a chord.

Proposition (Tangent and radius).

A tangent is perpendicular to the radius drawn to its point of contact, and two tangents drawn to a circle from the same external point are equal in length.

Theorem (Intersecting Chords Theorem).

If two chords of a circle cross at a point inside it, one cut into pieces of lengths $a$ and $b$ and the other into pieces of lengths $c$ and $d$, then $a \cdot b = c \cdot d$.

The radius to the point of contact is the shortest segment from the center to the tangent line, and the shortest segment always meets a line at a right angle. Two tangents from the same external point are then equal because each is a leg of a right triangle whose hypotenuse runs from the external point to the center and whose other leg is a radius.

Every chord through a given interior point yields the same product $a \cdot b$, so that number is called the power of the point.

Ways to work on it

• Walkthrough. Tangent-radius right angle, equal tangents, and crossing chords.
• Practice. Solve a crossing-chords product for the missing segment.
• Hardest. Tangent-secant power of a point.

Not sure where to start? Take the ten-question placement test.