Circles1294

=**//__Cirlces__//**=

__**Chords and Arcs**__




 * Circle -** A circle is the set of all points that are equidistant from a given point in the plane known as the center of the circle.


 * Radius -** A segment from the center of a circle to a point on the circle.


 * Chord -** A segment whose endpoints line on a circle.


 * Diameter -** A chord that contains the center of a circle.


 * Arc -** An unbroken part of a circle.


 * Endpoints -** Any two distinct point on a circle that divide the circle into two arcs.


 * Semicircle -** An arc whose endpoints are the endpoints of a diameter.


 * Minor arc -** An arc that is shorter than a semicircle of that circle.


 * Major arc -** An arc that is longer than a semicircle of that 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 and whose other points lie in the interior of the angle.


 * 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 degrees minus the degree measure of its minor arc. The degree meausre of a semicircle is 180 degrees.


 * 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 degrees (2//PIEr//)


 * Chords and Arcs Theorems -** In a circle, or in congruent circles, the arcs of congruent chords are equal.


 * __Tangents to Circles__



Seacant -** A line that intersects the circle at two points.


 * Tangent -** A line in the plane of a circle that intersects the circle at exactly one point.


 * Point of Tangency -** The point that the tangent intersects the circle at.


 * Tangent Theorem -** If a line is tangent to a circle, then the line is perpendicular to the radius of the cirle drawn to the point of tangency.


 * Radius and Chord Theorem -** A radius that is perpendicular that is perpendicular to a chord of a circle opposite the chord.


 * Convverse of 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.


 * __Inscribed Angles and Arcs__



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


 * Inscribed Angle Theorem -** The measure of an angle inscribed in a circle is equal to half the measure of the intercepted arc.


 * Right-Angle Corollary -** If an inscribed angle intercepts a semicircle then the angle is a right angle.


 * Arc-Intercept Collorary -** If two inscribed angles intercept the same arc, then they have the same measure.


 * __Angles Formed by Secants and Tangents__



Theorem -** If a tangent and a seacant (or chord) intersect on a circle at the point of tangency, then the measure of the angle formed is __half__ the intercepted arc.


 * Theorem -** The measure of an angle formed by two seacants or chords that intersect in the interior of a circle is half the sum of the measures of two arcs intercepted by the angle and it's vertical angle.


 * Theorem -** The measure of an angle formed by two seacants that intersect in the exterior of a circle.


 * __Segments of Secants, Tangents, and Chords__



Theorem -** If two segments are tangent to a circle from the same external point then the segments are perpendicular.


 * Theorem -** If two secants intersect ouside a circle, the product of the lengths of one secant segment and its external segment equals whole times outside. (WxO=WxO)


 * Theorem -** If a secant and a a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals tangent squared. (WxO=Tangent Squared)


 * Theorem -** If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the length of the other chord and segment.


 * __Circles in the Coordinate Plane__



Objective 1** - Develop and use the equation of a circle.


 * Objective 2 -** Adjust the equation for a circle to move the center in a coordinate plane.


 * __Examples__ - http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm**

-http://www.ies.co.jp/math/java/calc/limsec/limsec.html -http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_CircleSecantTangent.xml -[|http://www.ima.umn.edu/~arnold/calculus/secants/secants2/secants-g.html] -http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm -http://www.mathopenref.com/secantsintersecting.html
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