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A circle is the set of all points equidistant from a center point. The radius is the distance from the center to any point on the circle. The diameter is a chord passing through the center, equal to twice the radius. A chord is any segment with both endpoints on the circle. A secant is a line that intersects the circle at two points. These definitions are the foundation for all circle theorems.
Draw and label parts of a circle. Emphasize that a circle is a set of points (a curve), not the region inside it (that is a disk). Practice identifying radii, diameters, chords, and secants. Introduce the standard equation of a circle: (x-h)2 + (y-k)2 = r2.
A circle is defined by a single idea: equidistance. Pick a center point and a positive distance r. The circle is the set of all points in the plane that are exactly r units from the center — not closer, not farther. This makes a circle fundamentally different from a filled-in region (which is called a disk). The circle is the boundary curve alone.
From your prerequisite on segments and distance, you know how to measure the length between two points. The radius is the segment from the center to any point on the circle, and every radius of the same circle has the same length — that's the whole point of the equidistance definition. The diameter is a special chord: it passes through the center and connects two points on opposite sides of the circle. Because it consists of two radii laid end to end, the diameter is always exactly twice the radius: d = 2r. This relationship is worth memorizing because it appears in every circle formula you'll encounter.
A chord is any segment whose two endpoints both lie on the circle. The diameter is the longest possible chord — no chord can stretch farther than across the center. Every other chord is shorter, because straying from the center means the two endpoints are "closer together" along the circle. A secant generalizes the chord to a full line: where a chord is a segment that begins and ends on the circle, a secant is the infinite line that passes through those same two points, extending beyond the circle in both directions.
These definitions are not just vocabulary — they are the foundation for every theorem that follows. The circle's equation in coordinate geometry, (x − h)² + (y − k)² = r², is just the distance formula in disguise: any point (x, y) on a circle centered at (h, k) must satisfy the condition that its distance from the center equals r. When you see this equation later, recognize it as the definition of a circle restated algebraically. Every angle theorem, arc theorem, and tangent theorem you'll study builds directly on the precise meaning of radius, diameter, and chord established here.