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= = =__Chapter 9__=

**__Objectives__**

 * 1) Define a circle and its associated parts and use them in constructions.
 * 2) Define and use the de[[image:106820966_288a0268a8.jpg width="289" height="207" align="right" link="http://www.flickr.com/photos/geekmom/106820966/"]]gree measure of arcs.
 * 3) Define and use the length measure of arcs.
 * 4) Prove a theorem about chords and their intercepted arcs.

**__Definitions__**
__Circle__- 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__ is a segment from the center of the 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.

__Semicircle__- An arc whose endpoints are endpoints of a diameter.

__Minor arc__- Arc whose endpoints are endpoints of a diameter.

__Major arc__- Arc that is longer than a semicircle of that circle. Major Arc is named by its endpoints and another point that lies on the arc.

__Central Angle__- Angle in the plane ofa circle whose vertex is the center of the circle.

__Intercepted arc__- Inscribed angle that intersects. This place is called the intercepted arc.

__//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 of its minor arc. The degree measure ofa semicircle is 180.


 * Example**

Find the measures of arc CD, arc DB and arc CDB.

SOLUTION THe measures of CD and DB are found from their central angles.

mCD =115 mDB= 90

CD amd DB, which have an endpoint in common, are called adjacent arcs. Add their measures to find the measure of CDB.

m CDB =m CD + mDB= 115 + 90 = 205 degrees.

//9.2//
-Define tangents and **Tangent** secants of circles -Understand the relationship between tangents and certain radii of circles. -Understan the geometry of a radius perpendicular to a chord of a circle.
 * __Objectives__**


 * __Definitions__**
 * Secant** - a line that intersects with the circle at two points.

is a line in the plane of the circle that intersects the circle at exactly one point, which is known as the point of tangency. .
 * Tangent theorem -** If a line is tangent to a circle, then the line is 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 the chord.


 * Converse of the Tangent Theorem -** If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is to the circle.


 * Theorem -** The perpendicular bisector of a chord passes through the center of the circle.

**//9.3//**
- Define inscribed angle and intercepted arc. - Develop and use the inscribed angle theorem and its corollaries.
 * __Objectives__**


 * __Definitions__**
 * Inscribed angle theorem -** The measure of an angle inscribed in a circle is equal to 1/2 the measure of the intercepted arc.


 * Right Angle Corollary -** an inscribed angle intercepting the circle.


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


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

**//9.4//**
-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.
 * __Objectives__**


 * __Definitions__**
 * Theorem -** 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 1/2 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 1/2 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 1/2 the different of the measure of the intercepted arcs.

**//9.5//**
-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.
 * __Objectives__**


 * __Definitions__**


 * Tangent Segment -** a segment that is contained by a line tangent to a circle and has one of its endpoints on the circle.


 * Secant Segment -** a segment that contains a chord of a circle and has one endpoint exterior to the circle and the other endpoints on the circle.


 * External Secant Segment -** the portion of a secant segment that lies outside the circle.


 * Chord -** a segment whose endpoints lie on a circle.


 * Theorem -** If two segments are tangent to a circle from the same external points, then the segments are equal.


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


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


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

**//9.6//**
-Develop and use the equation of a circle. -Adjust the equation for a circle to move the center in a coordinate plane.
 * __Objectives__**


 * Example**

Given: //X//2 + //Y//2 = 25 Sketch and describe the graph by finding ordered pairs that satisfy the equation. Use a graphics calculator to varify your sketch.

SOLUTION When sketching the graph of a new type of equation, it is often helpful to locate the intercepts. To find the //x//-intercept(s), find the value(s) of //x// when y =0. (when a graph crosses the x-axis, //y//= 0.)

//x//2 + 02 = 49 //x//2 = 25 //x// = + or -7

Then the graph has tow //x// intercepts, (7,0) and (-7,0)

=**THE END**=