reca57

=CHAPTER 9= = = 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=radius black=diameter = = 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. = = Chords and Arcs Theorem In a circle, or in congruent circles, the arcs of congruent chords are congruent. The Converse of the Chords and Arcs Theorem In a circle or in congruent circles, the chords of congruent arcs are congruent. = = 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// = = //Blue Boxes// //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.//

//9.3//

Inscribed Angles and Arcs

Objectives = = Define incribed angle and intercepted arc. Develop and use the inscribed angle theorem and its corollaries. = = 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 = = http://flickr.com/photos/88261881@N00/365455942/

Objectives

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

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-half the sum 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-half the difference of the measures of the intercepted arcs The measure of a secant-tangent angle with its vertex outside the circle is one-half the difference. The measure of a tangent-tangent angle with its vertex outside the cirlce is one-half the difference of the measures 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.Theorems 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. ( Whole X Outside) = = 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.