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Circle Definitions

Parts Inside the Circle

A circle is the set of all points that area a fixed distance from a given point. That given point is called the circle's center.

A segment whose endpoints are both on the circle is called a chord. A diameter is a chord that passes through the center of the circle. A radius is a segment that has one endpoint on the circle and the other endpoint on the circle's center.

In the figure on the right you can see all of these points. O is the circle's center. segment AB is a chord. segment ED is both a chord and a diameter. segment OC is a radius.

circle with interior parts

Parts Outside the Circle

A line that intersects a circle in exactly one point is called a tangent. The point where the line intersects the circle is called the point of tangency. The segment between the point of tangency and another point on the tangent is called a tangent segment.

A line that intersects a circle in two points is called a secant. The segment from a point on the secant that's outside the circle to the farther of the two points where the secant and the circle cross is called a secant segment. The segment from a point on the secant to the closer of the two points where the line and the circle cross is called an external secant segment.

In the figure on the right, line AC is a tangent and B is its point of tangency. line EC is a secant. segment EC is a secant segment; segment CD is an external secant segment.

It would be incorrect to say that segment AC is a tangent segment. While it is a segment contained in a tangent, neither of its endpoints is the tangent's point of tangency.

circle with interior parts

Angles and Arcs

A central angle is an angle whose vertex is on the circle. An inscribed angle is an angle whose vertex is on the circle and whose sides contain chords of the circle.

An arc is the set of all points on the circle between two distinct points on the circle. Specifically, if O is the center of a circle and A and B are points on the circle then minor arc AB is the set of all points on the circle contained in ∠AOB, including A and B. Major arc AB is the set of all points outside of ∠AOB, including A and B. A semi-circle is an arc whose "central angle" is a diameter. It's usually defined as the set of all points on a circle that are on one side of a diameter.

In the figure on the right, ∠BDC is a central angle because its center is on the circle. ∠BAC is an inscribed angle because its vertex is on the circle and its sides contain chords of the circle.

arc BC is a minor arc because it is contained in ∠BDC. On the other hand, major arc BAC is a major arc because it is outside of central angle ∠BDC.

Arcs are measured in degrees just like angles are. Specifically, the measure of a minor arc is equal to the measure of its central angle. This means the measure of a semi-circle must be 180°. Because a circle is made up of two semi-circles, there must be 360° in a full circle. This means that the measure of a major arc is 360° minus the measure of its associated minor arc.

circle with interior parts

Postulates and Theorems

The Arc Addition Postulate If P is a point on arc AB them marc AB + marc PB = marc AB.
Theorem 1 The measure of an inscribed angle is equal to half the measure of its intercepted arc.
Theorem 2 All of the radii of a circle are congruent.
Theorem 3 In a circle if a radius is perpendicular to a chord then it bisects the chord.