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=Chapter 9= = = =~Section 9.1~=

Objectives
·Define a circle amd its associated parts and use them in construcions. ·Define ad use the degree measure of arcs. ·Define and ise the length measure of arcs. ·Prove a theorem about chords and their intercepted arcs.

Definition of Circle

 * Circle- a set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle.
 * Radius- a segment from the center of the circle to a point on te circle.
 * Chord- a segment whose endpoints line on the a circle.
 * Diameter- a chord that contains the center of a circle.

Definition of Centeral Angle & Intercepted Arc

 * Central Angle- an angle in the plane of a circle whose vertex is the center of the circle.
 * Intercepted Arc- with endopints that lie on the sides of the angle and the other points that lie in the interior of the angle.

Definition of Degree Measure of Arcs

 * Major Arc- 360° minus the degree measure of its minor arc.
 * Minor Arc- the measure f its central angle.
 * Degree Measure- of a semicircle is 180°

Arc Length
L=M/360°(2"Pie"r)
 * If r is the radius of a circle and m is the degree measure of an arc of the circle, then the length is L of the arc.

Chords and Arcs Theorem

 * In a circle, of in congruent circles, the arc f congruent chords are congruent.

The Converse of the Chords and Arc Theorem

 * In a circle or in congruent circles, the chords of congruent arcs are congruent.

=~Section 9.2~=

Objectives
·Define tangents and secants of circles. ·Understand the relationship between tangents and certain radii of circles. ·Understand the geometry of a radius perpendicular to a chord of a circle.

Definitions

 * Secant- to a circle is a line that intersects the circle at two points.
 * Tangent- is a line in the plane of the circle that intersects the circle at exactly one point.
 * Point of Tangency- Intersects the circle at one point.

Theorem

 * The perpendicular bisector of a chord passes through the center of the circle.

Tangent Theorem

 * If a line is tangent to a circle, than the line is perpendicular to a radius of the circle drawn to the point of tangency.

Radius and Chord Theorem

 * A radius that is perpendcular to a chord of a circle bisects the chord.

Converse of Tangent Theorem

 * If a line is perpendicular to a radius of a circle at its endpoint on the circle, than the line is tangent to the circle.

=~Section 9.3~=

Objectives
·Define inscibed angle and intercepted arc. ·Develop and use the Inscribed Angle Theorem and its corollaries.

Definition

 * Inscribed angle- An angle whose vertex lies on a circle and wose sides are chords of the circle.

Inscribed Angle Theorem

 * The measure of an angle inscribed in a circle is equal to one half the measure of the intercepted arc.

Right Angle Theorem

 * If an inscribed angle intercepts a semicircle, than the angle is a right angle.

Arc-Intercept Corollary

 * If two inscribed angles intercept that same arc, than they have the same measure.

=~Section 9.4~=

Objectives
·Define angles formed by secants and tangents of circles. ·Develop and use theorems about measures of arcs intercepted by these angles.

Theorem of Vertex On Circle

 * If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is one half the measure of its intercepted arc.

Theorem of Vertex Inside Circle

 * The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one half the sum of the measures of the arc intercepted by the angle and its vertical angle.

Theorem of Vertex Outside Circle

 * The measure of an angle formed by two secants that intersect in the exterior of a circle is one half the difference of the intercepted arcs.

=~Section 9.5~=

Objectives
·Define special cases of segments related to circles, including secant-secant, secant-tangent and chord-chord segments. ·Develop and ise theorems about measures of the segments.

Theorem Segments Formed by Tangent

 * If two segments are tangent to a circle from the same external point than the segments are of equal lengths.

Theorem Segments Formed by Secants
(Whole × Outside = Whole × Outside)
 * If two secants intersect outside a circle, tthe 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.


 * If a secant and a tangent intersect outside a circle, than the product of the lengths of the secant segment and its external equals the length of the tangent segment squared.

(Whole × Outside = Tangent Squared)

Theorem Segments Formed by Intersecting Chords

 * If two chords intersect inside a circle, than the product of the lengths of the segment of one chord equals the product of the length of the segment of the other chord

=~Section 9.6~=

Objectives
·Develop and use the equation of a circle. ·Adjust the equation for a circle to move the center in a coordinate plane.