chapter+nine

=9.1 Chords and Arcs=

Objectives
VOCABULARY [|Circle:]** A set of points in a plane that are equally distant from the center of the circle. Chords and Arcs therom,** 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 example, lets say you have a circle whith to radai in it and the angle of the two radai is 25 degrees. then the angle of the curve for those two is also going to be 25 degrees
 * 1.D**efine a circle and its assiated parts and use them in constructions
 * 2.** Define and use the degree meashure of arcs
 * 3.** Define and use the length measure of arcs
 * 4.** Prove a theorem about chords and their intercepthed arcs
 * [[image:gatwayarc.jpg width="221" height="122" link="http://flickr.com/photos/hometowninvasion/454859947/"]]
 * [|Radius:]** A segment that is connects the center of the circle to a point on the outside of the circle.
 * Chord:** A segment that extends from one point on the circles edge to another point on the circles edge.
 * D****iameter:** A segment that splits the circle in half by going through the center of the circle.
 * Semicircle:** Half of a circle.
 * Minor Arc:** An arc that measures smaller than 180°.
 * Major Arc:** An arc that measures greater than 180°.
 * Central Angle:** An angle with a vertex at the center of the circle and two endpoints on the edge of the circle.
 * Intercepted Arc:** The arc of a central angle.
 * THEOREM
 * Converse of the Chords and Arcs Theorem,**

=9.2 Tangents to Circles=

Objectives

 * 1.** Define tangents and secants of circles
 * 2.** Understant the relationship between tangents and certian radii of circles
 * 3.** Understand the geometry of a radius perpendicular to a chord of a circl

Secants and Tangents Theroem,** 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, wich is known as the point of tangency.
 * VACABULARY**
 * Secant:** A line that goes through the circle and intersects at two points.
 * Tangent:** A line that goes through the circle and intersects at one point.
 * Point of Tangency:** The point at which a tangent intersects.
 * Theorems
 * Radius and Chord Theorem,** A radius that is perpendicualr to ac chord of a circle bisects the chord.
 * 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.
 * Converse of the Tangent Theorem,** If a line is perpendicular to a radius of a circle at its endpoint o n the circle then the line is tangent to the circle.
 * Theroem,** The perpendicular bisector of a chord passes through the center of the circle.

=9.3 Inscribed Angles and Arcs=

Objectives

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


 * VOCABULARY**
 * Inscribed angle,** an angle whose vertex lies on a circle and whose sides are chords of the circle

Theorems,
example, with an inscribed angle you take the angle measure and multiply it by two to get the arc angle, or vise versa. =9.4 Angles Formed by Secants and Tgangents=
 * Inscribed angle theorem,** The measure of an angle inscribed in a circle is equal to one 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 Corollary,**If two insribed angles interrcept the same arc, then they have the same measure.

[|Objectives]

 * 1.** Define angles formed by secants and tangents of circles.
 * 2.** Decelop and use theorems about mearsures of arcs intercepted by these angles.

Theorems,

 * Theorem,** If a tangent and a secant(or a chord0 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.
 * 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.
 * 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 the intercepted arcs.
 * Theorem,** the measure of a scant tangent angle with its vertex outside the circle is one half the difference of the measujres of the intercepted arcs.
 * 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 degrees.

=9.5 Segments of Tangents, Secants,and Chords.=

[|Objectives]
this was made by me
 * 1.** Define a special cases of sefments ralated to circles includin secant secant, secant tangent, and chord chord segments
 * 2.** Develop and use theorems about measures of the segments.

Theorems,

 * Theorem,** If there are two tangents from the same external point, then the segments have an equal length.
 * Theorem,** If two secants intersected outside a circle, then the product of the whole segment and the outside segment equals the product of the whole segment and the outside segment.(Whole × outside=Whole × outside)
 * Theorem,** If two chords intersect inside the circle, then the product of the whole segment and the outside segment equals the length of tangent segment squared. ( Whole × outside=Tangent squared)
 * Theorem,** If two chords intersect inside a circle, then the product of part one and part two on one chord equal the product of part one and part two of the other chord. (Part 1 × Part 2 = Part 1 × Part 2)

=9.6 circles in the coordinate plane=

objectives

 * 1.** developand use the equation of a circle.
 * 2.** adjust the equation for a circle to move the center in a cordinate plane.