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=//**__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=**// The set of points in a plane that are equidistant from a given point known as the center of the circle.

//**Chord=**// A segment whose endpoins lie on a circle. //**Endpoints=**// A point at an end of a segment or the starting point of a ray. //**Minor arc=**// An arc of a circle that is shorter than a semi-circle of that circle. //**Major arc=**// An arc of a circle that is longer than a semi=circle of that circle
 * //Radius=//** A segment that connects the center of a circle with a point on the circle; one-half the diameter of a circle.
 * //Diameter=//** A chord that passes through the center of a circle; twice the length of the radius of a circle.[[image:bushelsfig2.jpg width="259" height="185"]]
 * //Arc=//** An unbroken part of a circle
 * //Semi-circle=//** The arc of a circle whose endpoints are the end points of a diamter.[[image:94.jpg]]

Central Angle - of a circle is an angle in the plane of a circle whose vertex is the center of the circle. Intercepted Arc- angle whose other points lie in the interior angle Degree Measure of Arcs- Of a minor arc- is the measure of its central angle Of a major arc- is 360 degrees minus the degree measure of its minor arc. Of a semi-circle- 180 degrees

Arc Length L=M/360 degrees (2

//**Intercepted arc=**// An arc whose endpoints lie on the sides of an inscribed angle. //**Degree measure of arcs=**// The measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360 minus the degree measure of it's central angle. //**Arc length=**// The measure of an arc in a circle in terms of degrees. //**The converse of the chords and Arcs Theorems=**// in a circle or in congruent circles, the arcs of congruent chords are

=//**__9.2 Tangents to Circles__**//=

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

//**Secant**//= To a circle is a line that intersects the circle at two points. //-The line AB is a tangent to the circle at P-// //**Tangent theorem=**// If a line is tangent to a circle, then the 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 intersects the chord. //**Converse of the tangent theorem=**// If a line is perpendicular to a radius of a circle at it's endpoint on the circle, then the line is parallel to the circle.
 * //Tangent=//** Is a line in the plane of the circle that intersects the circle at exactly one point, which is known as the point of tangency.

//**Theorem=**// The perpendiculat bisector of a chord that passes through the center of the circle.

=//**__9.3 Inscribed angles and arcs__**//=

Objectives- -Define inscribed angle and intercepted arc. -Develop and use the Inscribed Angle Theorem and its corollaries.

//**Inscribed angle theorem=**// is an angle whose vertex lies on a circle and whose sides are chords of the circle. //**Right-Angle corollary=**// If an inscribed angle intercepts a semicircle, then the angle is a right angle. //**Arc-intercept corollary=**// If two inscribed angles intercept at the same arc, then they have the same measure.

=**//__9.4 Angles formed by secants and tangents__//**=

//**Classification of angles with circles=**// Angles formed by pairs of lines that intersect a circle in two or more places can be studied systematically. There are three cases to consider, according to the placement of the vertex of the angles.

=__//9.5 Segments of tangents, secants, and chords//__=

//**Segments formed by tangents theorem**//= If two segments are tangent to a circle from the same external point, then the segments are of equal length.

//**Segments formed by secants theorem=**// If two secants intersect 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 it's external segment. (Whole x Outside= Whole x Outside) = = //**Segments formed by secants theorem part 2=**// If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared. //**Segments formed by intersecting chords theorem=**// 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 chord. = =