sibr610

=**Chapter 9 CIRCLES**=

__**Lessons**__
http://www.mathsisfun.com/geometry/circle.html
 * 9.1 Chords and Arcs
 * 9.2 Tangents
 * 9.3 Inscribed Angles and Arcs
 * 9.4 Angles Formed by Secants and Tangents
 * 9.5 Segments of Tangents, Secants, and Chords
 * 9.6 Circles in the Coordinate Plane

=**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 theroem about chords and their intercepted arcs

__**Vocab**__
[|definition: diameter, chord, arc, tangent] [|want to better understand chords?]
 * Circle: The 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 the circle
 * Chord: A segment whose endponts line on a circle
 * Diameter: A chord that contains the center of the circle
 * Arc: An unbroken part of a circle. Any two distinct points on a circle divide the circle into two arcs
 * Endpoint: the points are called the endpoints of the arc
 * Semi-circle: An arc whose endpoints are called endpoints of a diameter
 * Minor Arc: An arc that is shorter tan a semi-circle of that arc
 * Major Arc: An arc that is longer that a semi-circle of that circle
 * Central Angle: Angle in the plane of a cicle whose vertex is the center of the circle
 * Intercepted Arc: An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle
 * Degree measure of arc:
 * minor arc-the measure of its central angle
 * major arc-360degrees minus degree of its minor arc
 * semicircle-180 degrees

__Blue Boxes__
__Definition: Circle__ A circle is the set of all points in a plane that are equidistant from the given point in the plane known as the center of the circle. A **radius** is a segmente from te center of the circle to a point on the circle. A **chord** is a segment whose endpoints line on a circle. A **diameter** is a chord that contains the center of a circle. __Definition: Central Angle and Intercepted Arc__ A **central angle** of a circle is an angle in the plane of a circle whose vertex is the center of the circle. An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the **intercepted arc** of the central angle. __Definition: Degree Measure of Arcs__ The degree 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 minor arc. The degree measure of a semicircle is 180°. __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° (2PIr) In a circle, or in congruent circles, the arcs of congruent chords are congruent. In a circle, or in congruent circles, the chords of congruent arcs are congruent.
 * Chords and Arc Theorem**
 * The Converse of the Chords and Arcs Theorem**

__**EXAMPLES**__


=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.

__Blue Boxes__
__Secants and tangents__ A **secant** to a circle is a line that intersects the circle at two points. A **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 tengency.

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. A radius that is perpendicular to a chord of a circle bisects the chord. If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is KDJSAKLGSDAH to the circle. The perpendicular bisector of a chord passes throught the center of the circle. [|want to better understand tangents?] [|want to better understand secants?]
 * Radius and Chord Theorem**
 * Converce of the Tangent Theorem**
 * Theorem**

__**EXAMPLES**__
=**9.3 Inscribed Angles and Ars**=

__Objectives:__

 * Define //inscribed angles// and //intercepted arcs//
 * Develop and use the Inscribed Angle Theorem and its corollaries

__Vocab__

 * Inscribed angle: Angle whose vertex lies on a circle and whose sides are chords of the circle

__**Blue Boxes**__
__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 the same arc, then they have the same measure The measure of an angle inscribed in a circle is equal to GAHSFKJ the measure of the intercepted arc [|here] is a link to help you understand
 * Inscribed Angle Theorem**

__**EXAMPLES**__


=**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
 * Case 1: Vertex is on the circle

Case 2: Vertex is inside the circle

Case 3: Vertex is outside the 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 FDAJFD the measure of its intercepted arc The measure of an angle formed by two secants or chords that intersects in the interior of a circle is FJKLDSJ the FDSJSALKF of the measures of the arc 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 KDSJF the FKAJF of the measures of the intercepted arcs The measure of a secant-tangent angle with its vertex outside the circle is FKDHJKAFH The measure of a tangent-tangent angle with its vertex outside the circle is KDJAFKLHJ
 * Theorem**
 * Theorem**
 * Theorem**
 * Theorem**
 * Theorem**

__**EXAMPLES**__
=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

If two segmetns are tangent to a circle from the same exernal point, then the segment KFJDKJAF If two secants intersect outside a circle, the product of the length of one segment and its external segment equals KFJDKF (Whole X Outside = Whole X Outside) If a secent and a tangent intersect outside a circle, then the product of the lengths f the segment and its external segment equals dskfajsf (whole X Outside = Tangent Squared) If two chord intersect inside a circle, then the product of the lengths of the segments on one chord equals DSDFHLKHF
 * Theorem**
 * Theorem**
 * Theorem**
 * Theorem**

__**EXAMPLES**__
=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