Y. Davis Geometry Notes Chapter 10

Circle The locus or set of all points in a plane equidistant from a given point. The given point is called the Center. The center is the point that gives the circle its name.

Radius (Radii) A segment with one endpoint at the center and the other endpoint on the circle.

Chord A segment with both endpoints on the circle.

Diameter A chord that passes through the center of a circle. (Radius = ½ Diameter)

Concentric Circles Two or more circles with the same center, but different radii.

Circumference The distance around the circle. Pi

Inscribed/Circumscribed A polygon is inscribed in a circle if all of its vertices are points on the circle. A Circle is circumscribed about a polygon if it contains all the vertices of the polygon.

Central Angle An angle with its vertex at the center of a circle, its sides are radii of the circle.

Sum of Central Angles The sum of the central angles of a circle with no interior points is 360 degrees. A circle has 360 degrees.

Arc Part of a circle that is defined by the endpoints of 2 sides of a central angle. Arcs are congruent to the central angles that intercept them.

Minor Arc The shortest arc connecting two endpoints on a circle. Minor arcs measure less than 180 degrees

Major Arcs The largest arc connecting two endpoints on a circle Major arcs measure more than 180 and less than 360 degrees.

Semicircles An arc with endpoints that lie on the diameter of a circle Semicircles measure 180 degrees

Congruent arcs Arcs in the same or congruent circles that have the same measure

Theorem 10.1 In the same circle or congruent circle to minor arcs are congruent if and only if their corresponding central angles are congruent.

Adjacent Arcs Two arcs that have only one point in common.

Postulate 10.1 Arc Addition Postulate The measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs

Arc Length The distance between the endpoints along the arc measured in linear units.

Theorem 10.2 In the same circle are in congruent circles to minor arcs are congruent if and only if its corresponding chords are congruent.

Theorem 10.3 If a diameter (or radius) of a circle is perpendicular to a chord, then it bisects the chord and its arc.

Theorem 10.4 The perpendicular bisector of a chord is a diameter (or radius) of a circle.

Theorem 10.5 In the same circle or in congruent circles to chords are congruent if and only if they are equidistant from the center.

Inscribed Angles An angle with its vertex on the circle and its sides are chords of the circle.

Intercepted arc An arc that has endpoints on the sides of inscribed angle and lies in the interior of the inscribed angle.

Theorem 10.6 Inscribed Angle Theorem If an angle is inscribed in a circle, then the measure of the angle equals one half the measure of its intercepted arc.

Theorem 10.7 If two inscribed angles of a circle intercept the same arc or congruent arcs, then the angles are congruent.

Theorem 10.8 An inscribe angle of a triangle intercepts a diameter or semicircle if and only if the angle is a right angle.

Theorem 10.9 If a quadrilateral is inscribed in a circle, then its opposite angles are supplementary.

Tangent A coplanar line with a circle, that intersects the circle in exactly one point. (point of tangency)

Common Tangent A line, ray, or segment that is tangent to 2 coplanar circles. Common Internal Common External

Theorem 10.10 In a plane, a line is tangent to a circle if and only if it is perpendicular to a radius drawn to the point of tangency.

Theorem 10.11 If two segments from the same exterior point are tangent to a circle, then they are congruent.

Secant A line that intersects a circle in exactly 2 points.

Theorem 10.12 If 2 secants or chords intersect inside the circle, then the measure of an angle formed is one half the sum of the measure of the arcs intercepted by the angle and its vertical angle.

Theorem 10.13 If a secant and a tangent intersect at the point of tangency, then the measure of each angle formed is one half the measure of its intercepted arc.

Theorem 10.14 If 2 secants, a secant and tangent, or 2 tangents intersect outside the circle, then the measure of the angle formed is one half the difference of the measures of the intercepted arcs.

Summary Location of vertex Formula On circle Inside circle Outside Circle

Chord Segments When 2 chords intersect inside the circle they are divided into segments called chord segments.

Theorem 10.15 Segments of Chords Theorem If 2 chords intersect in a circle, then the products of the lengths of the chord segments are equal.

Secant Segment A segment of a secant line that has exactly one endpoint on the circle.

External secant segment A secant segment that lies in the exterior of the circle.

Theorem 10.16 If 2 secants intersect in the exterior of a circle, then the product of the measures of one secant segment and its external secant segment is equal to the product of the measures of the other secant and its external secant segment.

Tangent Segment A segment of a tangent with one endpoint on the circle

Theorem 10.17 If a tangent and a secant intersect in the exterior of a circle, ten the square of the measure of the tangent is equal to the product of the measures of the secant and its external secant segment.

Standard Form for the Equation of a Circle.