The many parts of a circle include arcs, tangents, segments, and chords. All these parts have various properties and theorems associated with them.
Circle: The set of all points in a plane that are equidistant (same distance away) from a specific point called the center.
Radius: (radii, plural) A line segment from the center to any point on the circle.
Diameter: A line segment from one point on the circle to another that contains the center of the circle.
Tangent: A line, line segment, or ray that intersects a circle at exactly one point.
Point of Tangency: The point where the tangent touches the circle.
Chord: A line segment whose endpoints are on a circle.
Secant: A line that intersects a circle in two points.
Arc: A section of a circle.
Semicircle: An arc that measures 180°.
Major Arc: An arc larger than a semicircle.
Minor Arc: An arc smaller than a semicircle.
Central Angle: An angle formed between two radii of a circle with its vertex at the center.
The center of a circle is a point, so the center is usually labeled with a capital letter like a point. The circle below is circle A, labeled \(\odot\)A.
Although the position of the center and the length of the radius may differ, all circles are similar to each other.
For two coplanar circles (two circles lying in the same plane), the circles can intersect in two points, one point, or no points.
Tangent circles are two circles that intersect in one point.
Circles that are not tangent can share a tangent line called a common tangent. The common tangent can also be internally or externally tangent.
Tangent to a Circle Theorem: A line is tangent to a circle if and only if the line is perpendicular to the radius drawn to the point of tangency.
Arcs are labeled with \(\frown\) The letters used to label an arc are the points on the circle
There are three types of arcs:
The central angle divides a circle into two arcs: either two semicircles or one major arc and one minor arc.
Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is the sum of measures of the two arcs
The endpoints of a chord lie on the circle, so a chord divides into two arcs.
Standard equation of a circle: \((x - h)^2 + (y - k)^2 = r^2\)
Since it’s (x - h) in the equation, remember to take the opposite sign of the value in the equation when finding h and k
To graph a circle:
To write the equation for a circle: