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Chapter 9 -Define a circle and its associated parts, and use them in constructions. -Define and use the degree of measure of arcse set of points in -Define and use the degree measure of arcs. -Prove a therem abought chords and their intercepted arcs.
 * Section 9.1**
 * Objectives**

-CentralTangle: An angle formed by tworays originatibng from the center of a circle. -Intercepted arc: An ark whose endpointslie on the sides of an inscribed angle. -Circle- A 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- A segment from the center of the circle to a point on te circle. -Chord- A segment whose endpoints line on the a circle. -Diameter- A chord that contains the center of a circle. -Definition of Centeral Angle & Intercepted Arc -Definition of Degree Measure of Arcs -Arc Length -Chords and Arcs Theorem 1.**Find the measures of arcs RT, TS and RTS. Arc mTS= 90 degrees.
 * Definitions**
 * Central Angle- an angle in the plane of a circle whose vertex is the center of the circle.
 * Intercepted Arc- with endopints that lie on the sides of the angle and the other points that lie in the interior of the angle.
 * Major Arc- 360° minus the degree measure of its minor arc.
 * Minor Arc- the measure f its central angle.
 * Degree Measure- of a semicircle is 180°
 * L=M/360°(2"Pie"r)
 * If r is the radius of a circle and m is the degree measure of an arc of the circle, then the length is L of the arc.
 * In a circle, of in cong
 * __Example One:__
 * 2.**The measures of arc RT and arc TS are found from their central angles.
 * 3.** Arc mRT = 100 degrees.
 * 4.** Arc RT and TS are adjacent angles. add their measures together to find the measure of arc RTS.

measure of arc RTS = measure of arc MRT+ MTS =100 degrees + 90 degrees. **190 Degrees.** Find length of the arc r =170mm length= 1/20 of the circumference of the circle. C=2*pi*r Length of arc: 1/20(2pi x 170) =17pi, approx. **53 mm**
 * __Example Two:__**

-Define tangenta and secants of circles. -Understand the relationship between tangents and certain radii of circles Tangents to Circles** (picture of circle) Find radius. r = 5 -Theorem -Tangent Theorem -Radius and Chord Theorem -Converse of Tangent Theorem -The Converse of the Chords and Arc Theorem
 * Section 9.2**
 * Objectives**
 * examples
 * Definitions**
 * Secant- to a circle is a line that intersects the circle at two points.
 * Tangent- is a line in the plane of the circle that intersects the circle at exactly one point.
 * Point of Tangency- Intersects the circle at one point.
 * The perpendicular bisector of a chord passes through the center of the circle.
 * If a line is tangent to a circle, than the line is perpendicular to a radius of the circle drawn to the point of tangency.
 * A radius that is perpendcular to a chord of a circle bisects the chord.
 * If a line is perpendicular to a radius of a circle at its endpoint on the circle, than the line is tangent to the circle.ruent circles, the arc f -congruent chords are congruent.
 * In a circle or in congruent circles, the chords of congruent arcs are congruent.


 * Section 9.3**

Objectives
·Define inscibed angle and intercepted arc. ·Develop and use the Inscribed Angle Theorem and its corollaries. (picture of circle) Find the following: mAB, m<ACD, m<AEB, m<BDP and m<P Answers: mAB = 100 m<ACD = 60 m<AEB = 100 m<BDP = 20 m<P = 40 Inscribed Angle Theorem Right-Angle Corollar Arc-Intercept Corollary
 * Examples**
 * Definition**
 * Inscribled angle- An angle whose vertex lies on a circle and wose sides are chords of the circle.
 * The measure of an angle inscribed in a circle is equal to one half the measure of the intercepted arc.
 * If an inscribed angle intercepts a semicircle, than the angle is a right angle.
 * If two inscribed angles intercept that same arc, than they have the same measure.


 * Section 9.4**

Objectives
·Define angles formed by secants and tangents of circles. ·Develop and use theorems about measures of arcs intercepted by these angles. -Theorem of Vertex On Circle -Theorem of Vertex Inside Circle -Theorem of Vertex Outside Circle
 * 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 arc 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 intercepted arcs.
 * Find mAVC in each figure...

A.** Angle AVC is formed by a secant and a tangent that intersect on the circle. By theorem 9.3.1, mAVC= ½mAV½ (150°)...
 * = 75°**

By theorem 9.3.2, mAVC=½ (mAC + mBD)... =½ (80° + 40°)...
 * B.** Angle AVC is formed by two secants that intersect inside the circle.
 * = 60°**

By theorem 9.3.3, mAVC=½ (mAC - mBD)... ½ (80° - 20°)...
 * C.** Angle AVC is formed by two secants that intersect outside the circle.
 * = 30°**


 * Section 9.5**

Objectives
·Define special cases of segments related to circles, including secant-secant, secant-tangent and chord-chord segments. ·Develop and ise theorems about measures of the segments. -Theorem Segments Formed by Tangent -Theorem Segments Formed by Secants (Whole × Outside = Whole × Outside)
 * If two segments are tangent to a circle from the same external point than the segments are of equal lengths.
 * If two secants intersect outside a circle, tthe product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment.

(Whole × Outside = Tangent Squared) -Theorem Segments Formed by Intersecting Chords Global positioning satellites are used in navigation. If the range of the satellite, AX, is 16,000 miles, what is BX?
 * If a secant and a tangent intersect outside a circle, than the product of the lengths of the secant segment and its external equals the length of the tangent segment squared.
 * If two chords intersect inside a circle, than the product of the lengths of the segment of one chord equals the product of the length of the segment of the other chord.
 * __Example One__:**

AX and BX are tangents to a circle from the same external point. By Theorem 9.5.1, they are equal. __**Example Two:**__ In the figure, EX=1.31 GX =0.45, and FX = 1.46. Find HX. Round your answer to the nearest hundredth.
 * __Solution:__**
 * AX= BX =16,000 miles.**

EX and FX are secants that intersect outside the circle. By Theorem 9.5.2, Whole x Outside = Whole x Outside.
 * __Solution:__**

EX x GX = FX x HX 1.31 x 0.45 = 1.46 x HX 1.46 x HX = 0.5895
 * HX = 0.40**

-Objectives·Develop and use the equation of a circle. ·Adjust the equation for a circle to move the center in a coordinate plane.
 * Section 9.6**
 * __Equations:__**
 * 1.** When the center of the circle is at the origin (0,0)
 * X² + Y² = r ²**

(h,k) is the origin. 1.** A circle on the origin (0,0) and a radius of 5 x^2 + Y^2 = 5^2 (x - 5)^2 + (y - 4)^2 = 3^2
 * 2.** When the center of the circle is not at the origin.
 * (X - h)² + (Y - k)² = r ²**
 * __Example:__
 * 2.** A circle with a center of (5,4) and a radius of 3

other web sites to try http://www.geom.uiuc.edu/%7Edwiggins/conj44.html [|If help is needed try this web site] http://library.thinkquest.org/20991/geo/circles.html