Chapter 10: Circles Review

Vocabulary and theorems regarding circle geometry, including properties of tangents, arcs, chords, inscribed angles, and the standard equation of a circle.

Circle

The set of all points in a plane that are equidistant from a given point called the center of the circle.

Radius

A segment whose endpoints are the center and any point on a circle.

Chord

A segment whose endpoints are on a circle.

Diameter

A chord that contains the center of the circle.

Secant

A line that intersects a circle in two points.

Tangent

A line in the plane of a circle that intersects the circle in exactly one point, known as the point of tangency.

Tangent Line to Circle Theorem

In a plane, a line is tangent to a circle if and only if the line is perpendicular to a radius of the circle at its endpoint on the circle.

External Tangent Congruence Theorem

Tangent segments from a common external point are congruent.

Arc

Part of a circle between two points on the circle.

Minor Arc

An arc that measures less than $180$ degrees.

Major Arc

An arc that measures more than $180$ degrees.

Semi-circle

An arc that measures exactly $180$ degrees.

Central Angle

An angle of a circle that has its vertex at the center and its sides are radii; its measure is equal to the measure of the intercepted arc.

Adjacent Arcs

Two arcs of the same circle that intersect at exactly one point.

Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs.

Congruent Circles Theorem

Two circles are congruent circles if and only if they have the same radius.

Congruent Central Angles Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding central angles are congruent.

Similar Circles Theorem

All circles are similar.

Congruent Corresponding Chords Theorem

In the same circle, or in congruent circles, two minor arcs are congruent if and only if their corresponding chords are congruent.

Perpendicular Chord Bisector Theorem

If a diameter of a circle is perpendicular to a chord, then the diameter bisects the chord and its arc.

Perpendicular Chord Bisector Converse

If one chord of a circle is a perpendicular bisector of another chord, then the first chord is a diameter.

Equidistant Chords Theorem

In the same circle, or in congruent circles, two chords are congruent if and only if they are equidistant from the center.

Inscribed Angle

An angle of a circle that has its vertex on the circle and the sides are chords of the circle.

Measure of an Inscribed Angle Theorem

The measure of an inscribed angle is one-half the measure of its intercepted arc.

Inscribed Angles of a Circle Theorem

If two inscribed angles of a circle intercept the same arc, then the angles are congruent.

Inscribed Polygon

A polygon where all its vertices lie on a circle.

Circumscribed Circle

The circle that contains the vertices of an inscribed polygon.

Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse is a diameter of the circle; conversely, if one side is a diameter, the triangle is a right triangle.

Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if its opposite angles are supplementary.

Tangent and Intersected Chord Theorem

If a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one-half the measure of its intercepted arc.

Angles Inside the Circle Theorem

If two chords intersect inside a circle, the measure of each angle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Angles Outside the Circle Theorem

If a tangent and a secant, two tangents, or two secants intersect outside a circle, the measure of the angle is one-half the difference of the measures of the intercepted arcs.

Circumscribed Angle

An angle whose sides are tangent to a circle.

Circumscribed Angle Theorem

The measure of a circumscribed angle is equal to $180$ degrees minus the measure of the central angle that intercepts the same arc.

Segments of Chords Theorem

If two chords intersect in the interior of a circle, then the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord.

Tangent Segment

A segment that is tangent to a circle at an endpoint.

Secant Segment

A segment that contains a chord of a circle and has exactly one endpoint outside the circle.

External Segment

The part of a secant segment that is outside the circle.

Segments of Secants Theorem

If two secant segments share the same endpoint outside a circle, the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

Segments of Secants and Tangents Theorem

The product of the lengths of the secant segment and its external segment equals the square of the length of the tangent segment.

Standard Equation of a Circle

The equation $(x - h)^2 + (y - k)^2 = r^2$ where $(h, k)$ is the center and $r$ is the radius.