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=Chapter 9 Circles=

9.1 Chords and Arcs
__Objectives:__ -Learn the definitions to a circle and its associated parts, and use them in constructions. -Learn the definition and use the degree measure of arcs. -Learn the definitions and use the length measure of arcs. -A theorem that is about chords and their intercepted arcs. __Definitions and theorms:__ //Circle-// A set of all points in a plane that are equidistant from a given point in the plane that is the center of the circle. //Radius-// A line from the edge of the circle to the center point of a circle. Chord-A segment whose endpoints go from one point on the edge of a circle to another point on the edge of a circle. //Diameter-// A segment that goes from edge to edge of a circle but that also goes through the center point of the circle. //Arc-// A part of a cicle that is not broke. //Endpoints-// Two points on the edge of a cicle that if there was a line connecting them it would divide the circle into two arcs. //Semicircle-// They arc of a diameter. //Minor arc-// Its an arc that is shorter than a semicircle. //Major arc-// Its an arc that is longer than a semicircle. //Centeral angle-// Angle in the plane of a circle which the vertex in the center of the circle. //Intercepted arc-// An arcs endpoints that lie on the sides of the angle and which the other points lie on the interior angle. //Degree measure of arcs-// A minor arc-measure of a centeral angle. //A major arc-// 360° minus the degree measure of its minor arc. 180° is the degree measure of a semicircle. //Arc length-// L=M/360°(2pir) M- Degree measure L- length r- radius //Chords and arcs theorem//- In circles the arcs of congruent chords are congruent The converse of the chords and arc theorem- In circles, chords of congruent arcs are congruent http://library.thinkquest.org/10030/13arcsandc.htm

9.2 Tangents to Circles
__Objectives:__ -Find the defintions for tangents and secants of circles. -Learn the relationship between tangents and certain radii of circles. -Learn the geometry of a radius perpendicular to a chord of a circle. __Definitions and theorems:__ //Secant-// It intersects a circle at two points //Tangent-// It intersects the edge of a circle at one point //Point of tangency-// The point where a tangent line intersects //Tangent Theorem-// A line that is tangent to a circle, is also perpendicular to a radius of the circle drawn to the point of tangency. //Radius and Chord Theorem//- Radiuses that are perpendicular to a chord in a circle bisectsthe chord. //Converse of the Tangent Theorem-// A line that is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle. //Theorem//- A perpendicular bisector of a chord passes through the center of a circle.

[|Secant and tangent circles] the whole page

9.3 Inscribed Angle and Arcs
__Objectives:__ -Learn the definitions for inscribed angle and intercepted arcs. -Learn and use the inscribed angle theorem and its corollaries. __Definitions and theorems:__ //Inscribed Angle Theorem-// A angle that si inscribed in a circle is equal to one-half the measure of the intercepted arc. //Right-Angle Corollary-// If it is a right angle then a inscribed angle intersepts a simicircle. //Arc-Intercept Corollary-// Two inscribed angles that would intercept the same arc, have the same measure.

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[|Inscribed angles] scroll down to about the middle till it says inscribed angles

9.4 Angles Formed by Secants and Tangent
__Objectives:__ - Definitions for angles formed by secants and tangents of circles. -Learn and use theorems about measures of arcs intercepted by these angles.

Theorms: //Vertex on circle-secant and tangent theorem-// If a secant and a tangent line intersect on a circle at the point of tangency, then the //measure of the angle formed is one//-half the measureof its intercepted arc. //Vertex inside circle-two secants theorem-//The masure of an angle formed by two secants or two chords that intersect in the interior part of a circle is one-half teh sum of the measures of the arcs intercepted by the angle and its vertiacl angle. //vertex outside circle-//two secants theorem- An angle with a measure formed by two secants that intersect the exterior part of a circle is one-half the difference of the measures of the intercepting arcs //Theorem #4-// A secant-tangent angle that has measures and also with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs. //Theorem #5-// A tangent-tangent angle with a measure and also with its vertex outside the circle is one-half the difference of the intercepted arcs, or the major arcs measure minus 180°. [|Angles] it all has something to do with this chapter

9.5 Segments of Tangents, Secants, and Chords
__Objectives:__ -Learn and use theorems about measures of segments. -Define special cases of segments related to circles, like secant-secant, secant-tangent, and chord-chord segment. __Theorems:__ //Segments formed by two tangents theorem-//Two segments that are tangent to a circle from the same external point, and the segments are equal lengths. //Segments formed by secants theore//m- Two secants that intersect outside a circxle, the product of the lengths of one secant segment adn the external segments quals the product of the lengths of th other secant segment and the external segments also. //Theorem #2 of segments formed by secants//- A secaant and a tangent that intersect outside a circle, and the product of the lengths of the secant segment and its external segment equals the length of the segment that is tangent squared. //Segment formed by intersecting chords theorem-// Two chords that intersect inside the circle, and the product of the lengths of the segments of one chords equals the product of the segments lengths of the other chord. [|secant-secant] picture and definition = =

9.6 Circles in the Coordinate Plane
__Objectives:__ - Learn and use the equation for a circle - Learn how to change the equation for a circle to move the center in a coordinate plane