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=**__Chapter 9__**=

__**9.1**__ //Chords and Arcs// A circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle. A **radius** ( plural, radii) is a segment from the center of the circle to a point on the circle. A **chord** is a segment whose endpoints line o a circle. A **diameter** is a chords that contains the center of a circle. An **arc** is an unbroken part of a circle. Ant two distinct points on a circle divide the circle into two arcs. The points are called the **endpoints** of the arcs. A semicircle is an arc whose endpoints are endpoints of a diameter. A semicircle is informally called a half-circle. A semicircle is named by it's endpoints and another point that lies on the arc. 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. 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. __**Degree Measures of Arcs**__ A **central angle** of a circle is an angle in the plane of circle whose vertex is the center of the circle. An arc whoes endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the **intercepted arc** of the central angle. 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 measure of a semicircle is 180 degrees. If r us the radius of a circle and M is the degree measure of an arc of the circle, then the length, of the arc is given by the following: L=M/360( 2x3.1xr) In a circle, or in congruent circles, the arcs of congruent chords are _ In a circle on in congruent circles, the chords of congruent arcs are_ Identify chords and their arcs the chord,** [|**minor arc**]**, the** [|**major arc**]**, the chord's arc.**
 * __Objectives:__**
 * Define a circle and its associated parts, and use them in construction.
 * 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.
 * __Definition: Circle__**
 * __Major and Minor Arcs:__**
 * Definitions: Central Angle and Intercepted Arc**
 * Definition: Degree Measured of Arc**
 * __Arc Length__**
 * __Chord and Arcs Theorem__**
 * __The Converse of the Chords and Arcs Theorem__**
 * __EXAMPLE__**
 * Identify chords and their arcs**
 * In the diagram on the left, identify:

[|More Deffinitions!!!!] [|Circles Are Awsome!!!] [|Practice Makes Perfect!!! Here are some more!]
 * __Fun " MATH " Sites To Visit__**

//Tangents and circles// 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 that circle at exactly one point, which is known as the point of tangency. If a line is tangent to a circle, then the line is to a radius of the circle drawn to the point of tangency. A radius that is perpendicular to a chord of a circle the chord. If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is to the circle. The perpendicular bisector of a chord passes through the center of the circle.
 * __9.2__**
 * __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.
 * __Secants and Tangents__**
 * __Tangent Theorem__**
 * Radius and Chord Theorem**
 * __Converse of the Tangent Theorem__**
 * Theorem**


 * __EXAMPLE__**
 * Use the** [|**theorem**] **above to find the measure of angle formed by the intersection of the tangent that intersects chord AC**. [[image:http://www.mathwarehouse.com/geometry/circle/images/angles-tangent-chord/picture-tangent-chord-angle.jpg width="326" height="277" align="left" link="http://http://www.mathwarehouse.com/geometry/circle/angle-tangent-and-chord.php"]]
 * Measure of angle??**

=½ • 170= =**85°**
 * By the [|theorem], the measure of angle is half of the [|intercepted arc] which is 170°. Therefore x**

__**Fun " MATH" Sites to visit**__ [|Tangents!!] [|Who wants to be a mathonaire?] [|More and More practice!!]

__**9.3** //Inscribed Angles and Arcs//

An **inscribed angle** is an angle whose vertex lies on a circle and whose sides are chords of the circle. The measure of an angle inscribed in a circle is equal to the measure of the intercepted arc. If an Inscribed angle intercepts a semicircle, then the angle is a right angle. If two inscribed angles intercept the same arc, then they have the same measure.
 * Objectives**__
 * Define inscribed angles and intercepted arc.
 * Develop and use the Inscribed Angle Theorem and its corollaries.
 * The Inscribed Angle Theorem**
 * Right-Angle Corollary**
 * Arc-Intercept Corollary**


 * __EXAMPLE__**
 * Practice** **Identifying the Inscribed Angles and their Intercepted Arcs**
 * Identify the inscribed angles and their intercepted arcs**

If XYZ Measure of inscribed angle ½ measure of the intercepted arc Therefore, = 2 × 40o80o

__**Fun " MATH" Sites to visit**__ [|Interactive Central Angle of a circle] [|You guessed it more Practice!]

__**9.4**__ __//Angles Formed by Secants and Tangents//__ > *There are a lot of theorems in this chapter!! __**Theorem**__ If a tangent and secant ( or a chord) intersect on a circle at the point of tangency, then the measure of the angle formed is the measure of its intercepted arc. The measure of an angle formed by two secants or chords that intersect in the interior of the circle is the of the measures of the arcs intercepted by the angle and its vertical angle. __**Theorem**__ The measure of the angle formed by two secants that intersect in the exterior of a circle is the of the measures of the intercepted arcs. __**Theorem**__ The measure of a secant-tangent angle with its vertex outside the circle is The measure of a tangent-tangent angle with its vertex outside the circle is
 * Objectives**
 * Define angles formed by secants and tangents of circles.
 * Develop and use theorems about measures of the arcs intercepted by these angles.
 * Classification of angles with Circles**
 * 1) Vertex is on the Circle.
 * 2) Vertex is inside the Circle.
 * 3) Vertex is outside the Circle.
 * __Theorem__**
 * Theorem**

Angle outside the circle formed by two secants**
 * __Example__

Below in figure b, DCP from the diagram shown in figure a above.

As shown in the diagram above, ÐDCP is supplementary to The three angles ofDCP must have a sum of 180°. Solving this for :def P yeilds This means that the measure of P, an external to and formed by two secants, is equal to one half the difference of the intercepted arcs.

__**9.5**__ //Segments of Tangents, Secants, and Chords//
 * Objectives**
 * Define a 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.

__**Theorem**__ If two segments are tangent to a circle from the same external point, then the segments_ __**Theorem**__ If two secants intersected outside the circle, the product of the lengths of one secant segment and its external segment equals_ ( Whole x Outside = Whole x Outside) __**Theorem**__ If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals ? ( Whole x Outside = Tangent Squared ) If two chord intersect inside a circle, then the product of the lengths of the segments of one chord equals_
 * __Theorem__**


 * __EXAMPLE__**
 * Use the theorem for the intersection of a tangent and a secant to find the measure of the angle formed by the intersection of the tangent and the secant.** [[image:http://www.mathwarehouse.com/geometry/circle/images/power-of-point/tangent-secant-picture.jpg width="350" height="233" align="left" link="http://http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php"]]

[|Formulas to Remeber] [|More Practice Problems!!!!]
 * __Fun "MATH" sites to visit__**

//Circles in the coordinate plane//
 * __9.6__**

Develop and use the equation of the circle Adjust the equation for a circle to move the center in a coordinate plane.b In your work in algebra, you may have investigates graphs of equations such as y=2x-3 ( a line ), y= x^2-3 ( a parabola ), and y=3 x 2^x ( an exponential curve). In this lesson, you will investigate equations in which both x and y are squared.
 * __Objective:__**
 * Graphing a circle from an Equation**

Given: X^2 + Y^2 = 25 Sketch and describe the graph by finding orderd pairs that satisfy the equation. Use a graphics calculator to verify 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 graph crosses the x-axiz, y=0 ) X^2 + 0^2 = 25 X^2= 25 X = + 5 Thus, the graph has two x- intercepts, ( 5,0) and (-5, 0)
 * __Example__**


 * It Could Be Worse!!!**