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=Chapter 9=

Objectives

 * 1) Define a circle and its associated parts, and use them in constructions
 * 2) Define and use the degree measure of arcs
 * 3) Define and use the length measure of arcs
 * 4) Prove a theorem about chords and their intercepted arcs

Definitions
This is an example of a semicircle ^
 * **Circle** - The set of points in a plane that are equidistant from a given point known as the center of the circle.
 * **Radius** - A segment that connects the center of a circle with a point on the circle; one-half the diamter of a circle.
 * **Chord** - A segment whose endoints lie on a circle.
 * **Diameter** - A chord that passes through the center of a circle; twice the length of the radius of the circle.
 * **Arc** - An unbroken part of a circle.
 * **Endpoints** - A point at the end of a segment or the starting point of a ray.
 * **Semi-circle** - The arc of a circle whose endpoints are the endpoints of a diameter
 * **Minor arc** - An arc of a circle that is shorter than a semicircle of that circle.
 * **Major arc** - An arc of a circle that is longer than a semicircle of that circle.
 * **Central angle -** An angle formed by two rays originating from the center of a cricle**.**
 * **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 its central angle.

Theorems

 * 1) **__Arc Length__** - If //r// is the radius of a circle and //M// is the degree measure of an arc of the circle, then the length, L, of the arc is given by the following: L = M/360°(2πr).
 * 2) **__Chords and Arcs Theorem__** - In a Circle, or in congruent circles, the arcs of congruent chords are congruent.
 * 3) **__Converse of Chords and Arcs Theorem__** - In a circle, or in congruent circles, the chords of congruent arcs are congruent.

Objectives

 * 1) Define tangents and secants of circles.
 * 2) Understand the relationship between tangents and certain radii of circles.
 * 3) Understand the geometry of a radius perpendicular to a chord of a circle.

Definitions

 * **Secant** - A line that intersects the circle at two points.
 * **Tangent** - A line in the plane of the circle that intersects the circle at exactly 1 point.
 * **Point of tangency** - The point of intersection of a circle or spherer with a tangent line or plane.

Theorems

 * 1) **__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.
 * 2) **__Radius and Chord Theorem__** - A radius that is perpendicular to a chord of a circle bisects the chord.
 * 3) **__Converse of the Tangent Theorem__ -** If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.
 * 4) **__Theorem__** - The perpendicular bisector of a chord passes through the center of the circle

Objectives

 * 1) Define inscribed angle and intercepted arc.
 * 2) develop and use the inscribed angle theorem and its corollaries.

Definitions

 * **Inscribed angle** - An angle whose vertex lies on a circle and whose sides are chords of the circle.
 * **Intercepted arc** - An arc whose endpoints lie on the sides of an inscribed angle.

Theorems and Corollaries

 * 1) **__Inscribed Angle Theorem__** - The measure of an angle inscribed in a circle is equal to one-half the measure of the intercepted arc.
 * 2) **__Right Angle Corollary__** - If an inscribed angle intercepts a semicirlce, then the angle is a right angle.
 * 3) **__Arc-Intercept Corollary__** - If two inscribed angles intercept the same arc, then they have the same measure.

Objectives

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

Theorems

 * 1) **__Theorem__** - 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.
 * 2) **__Theorem__** - The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.
 * 3) **__Theorem__** - The measure of a secant-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs.
 * 4) **__Theorem__** - 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 arcs intercepted by the angle and its vertical angle.
 * 5) **__Theorem__** - The measure of a tangent-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180°.

Objectives

 * 1) Define special cases of segments related to circles, including secant-secant, secant-tanget. and chord-chord segments.
 * 2) Develop and use theorems about measures of the segments

Theorems

 * 1) **__Theorem__** - If two segments are tangent to a circle from the same external point, then the segments are of equal length
 * 2) **__Theorem__** - If two secants intersect outside a circle, then 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. (Whole × Outside = Whole × Outside)
 * 3) **__Theorem__** - 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. (Whole × Outside = Tangent Squared [or t²] )
 * 4) **__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

Objectives

 * 1) Develop and use the equation of a circle.
 * 2) Adjust the equation for a circle to move the center in a coordinate plane.