Circle Vocabulary

Circle theorems are mostly vocabulary. Get the words right and the theorems become readable.

• Radius: centre to the edge.
• Chord: a segment with both endpoints on the circle.
• Diameter: a chord through the centre. The longest chord, and twice the radius.
• Secant: a *line* through two points of the circle.
• Tangent: a line touching at exactly one point.

The single most used fact: a tangent is perpendicular to the radius drawn to the point of contact. That right angle is what lets you bring in the Pythagorean theorem.

Two more worth knowing: two tangents drawn from the same outside point are congruent, and concentric circles share a centre.

A circle is the locus of points at a fixed distance r from a centre. Chords, secants and tangents are classified by how many points they share with that locus: two, two, and one respectively.

The Tangent-Radius Theorem states that a tangent is perpendicular to the radius at the point of tangency, and its converse holds — a line perpendicular to a radius at its endpoint on the circle is tangent. It is proved indirectly: any other point of the tangent line lies outside the circle, so the radius is the shortest distance and therefore perpendicular.

The Two Tangents Theorem states that tangent segments from a common external point are congruent, proved by HL on the two right triangles formed with the radii.

The perpendicularity is what makes circle problems computable, since it introduces right triangles with the radius as one leg and the tangent as the other, letting the Pythagorean theorem relate the tangent length to the distance from the centre.

A polygon is inscribed in a circle when all vertices lie on it, and circumscribed about a circle when all sides are tangent to it.