RaKa691

=__**9.1 - Chords and Arcs**__=


 * __Objectives__**
 * Define a circle and its associated parts and use them in constructions.
 * Define and use the degree measure of arcs.
 * Define and use the length measure of arcs.
 * Prove a theorem about chords and their intercepted arcs.


 * Circle -** All points in a plane that are the same distance from a point (the center of the circle)
 * Radius** - A segment from the center of the circle to a point on a circle
 * Chord** - A segment whose endpoints line on a circle.
 * Diameter** - A chord that contains the center of a circle.
 * Central Angle** - An angle in the plane of a circle whose vertex is the center of the circle.
 * Intercepted Arc** - An arc whose endpoints lie on the sides of the angle.

__**Blue boxes**__ : An __arc__ is an unbroken part of the circle. Any two distinct points [called enpoints] on a circle divide the circle into two arcs. A __semicircle__ is an arc whose endpoints are endpoints of a diameter. The measure of a semicircle is 180 degrees. A __minor arc__ of a circle is an arc that is shorter than a semicircle of that circle. A minor arc is named by its endpoints. The degree measure of a minor arc is the measure of its central angle. A __major arc__ of a circle is an arc that is longer than a semicircle of that circle. A major arc is named by its endpoints and another point that lies on the arc. The degree measure of a major arc is 360 minus the degree measure of its minor arc.

In a circle, or in congruent circles, the arcs of congruent chords are __congruent__.
 * Chords and Arcs Theorem**

In a circle or in congruent circles, the chords of congruent arcs are __congruent__.
 * The Converse of the Chords and Arcs Theorem**

Arc Length //R// is the radius of a circle, //M// is the degree measure of an arc of the circle and //L// is the length, the arc is given by: L = M / 360° (2 x Pi x R)

=__9.2 - Tangents to Circles__=


 * __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.


 * Secant**- Line that intersects the circle at two points
 * Tangent-** Line in theplane of the circle that intersects the circle at exactly one point
 * Point of tangency-** The point a tangent intersects a circle

Tangent theorem: If a line is tangent to a circle then te line is perpendicular to a radius of the circle drawn to the point of tangency. Radius and chord theorem: A radius that is perpendicular to a chord of a circle bisects the chord. Converse of the tangent theorem: If a line is perpendicular to a radius of a circle at its endpoint on circle then the line is tangent to the circle Theorem: The perpendicular bisector of a cord just passes through the center of the circle.
 * __Blue Boxes__** :

=__9.3 - Inscribed Angles and Arcs__=


 * __Objectives__**
 * Define incribed angle and intercepted arc.
 * Develop and use the inscribed angle theorem and its corollaries.
 * Inscribed Angle** - An angle whose vertex lies on a circle and whose sides are chords of the circle.


 * __Blue Boxes__**
 * Inscribed Angle Theorem:** The measure of an inscribed is equal to half the measure of the intercepted arc.
 * Right Angle Corollary:** If an inscribed angle intercepts a simicircle, then the angle is a right angle.
 * Arc-Intercepted Corollary:** If two inscribed angles intercept the same arc, then they have the same measure.

=__**9.4 - Angles Formed By Secants and Tangents**__=


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


 * __Blue Boxes__**
 * 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.
 * The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-halfone-half the sumsum of the measures of the arcs intercepted by the angle and its vertical angle.
 * The measure of an angle formed by two secants that intersect in the exterior of a circle is one-halfone-half the differencedifference of the measures of the intercepted arcs
 * The measure of a secant-tangent angle with its vertex outside the circle is one-half the differenceone-half the difference.
 * The measure of a tangent-tangent angle with its vertex outside the cirlce is one-half the difference of theone-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180measures of the intercepted arcs, or the measure of the major arc minus 180°

=**__9.5 - Segments of Tangents, Secants, and Chords__**=


 * __Objectives__**
 * Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments
 * Develop and use theorems about measures of the segments.


 * __Blue Boxes__**
 * If two segments are tangent to a circle from the same external point, then the segments are of equal length.
 * If two secants intersect outside a circle, the product of the lenghts of one secant and its external segment equals the product of the lengths of the other secant segment and its external segment product of the lengths of the other secant segment and its external segment.
 * -( Whole X Outside = Whole X Outside )
 * If a secant and a tangent intersect outside a circle then the product of the lingths of the secand sgmet and its external segment equals the tangent squared
 * -(whole X outside=Tangent Squared)'
 * if two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other.

=__9.6 Circles in the coordinate plane__=
 * __Objectives__**
 * Develop and use the equation of a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.

The equation for a circle is x²+y²=r²