brtr57


 * __Chapter 9.1__**

__Objective:__ -Define a circle and its associated parts, and use them in constructions -Define and use the degree measure of arcs. -Define and use the length measure of arcs. -Prove a theorem about chords and use their intercepted arcs.

__Definitions:__ -Circle: The set of all points in a plane that are equidistant from a given point in the plane known as center of circle. -Radius: segment from the center -Chord: segment whose endpoints line on a circle. -Diameter: A chord that contains the center of a circle. -Arc: An unbroken part of a circle. -Endpoints: Two distinct points on a circle divide the circle into two arcs. -Semi-circle: An arc whose endpoints are endpoints of a diameter. -Minor arc: A circle is an arc that is shorter than a semi-circle of that circle. -Major arc: An arc that’s longer than a semi-circle of that circle named by endpoints and another point that lies on the arc. -Central angle: A circle is an angle in the plane of a circle whose vertex is the center of a circle. -Intercepted arc: An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle. -Degree measure of arcs: The degree measure of a minor arc is the measure of its central angle.

__Theorems:__ -Arc length: If r is the radius of a circle and M is the degree measure of an arc of the circle, then the length, L, of the arc is shown here: L=M/360(2pir) -Chords and Arcs Theorems: In a circle, or in congruent circles the arcs of congruent chords are the same.

__Example:__ The wheel of a bicycle has a radius of 20 in. and the degree of one of it's arcs is 20 degrees. Find the length. L=20/360(2pi20) L=6.28

http://www.worsleyschool.net/science/files/sector/calculations.html


 * __Chapter 9.2__**

__Objectives:__ -Define tangents and secants of a circle. -Understand the relationship between tangents and certain radii of circles. -Understand the geometry of a radius perpendicular to a chord of a cicle.

__Definitions:__ -Secant: A line that intersects the circle at two points. -Tangent: A line in the plane of the cicle that intersects the circle at exactly one point. -Point of Tangency: Point where tangent intersects the circle.

__Theorems:__ -Tangent theorem: If a line is tangent to a circle, then the line is tangent to a radius of the circle drawn to the point. -Radius and Chord theorem: A radius that is perpendicular to a chord of a circle is equal to the chord. -Converse of tangent theorem: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is congruent to the circle.

__Example:__ Imagine a circle with a two perpendicular lines intersecting at its center. Half of one of the lines measures 50 in. and the intersecting half of the other line measures 45 in. If you drew a line from the top point of one line to the side point of the intersecting line than it would form a right triangle. Find the measure of that chord with pathagorean theorem: 50^2+45^2=?^2 chord= 67.27

http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm


 * __Chapter 9.3__**

__Objectives:__ -Define inscribed angle and intercepted arc. -Develop and use the Inscribed Angle Theorem and its corollaries.

__Definitions:__ -Inscribed angle: is and angle whose vertex lies on a circle and whose vertex lies on a circle and whose sides are chords of the circle.

__Theorems:__ -Inscribed angle theorem: The measure of an angle inscribed in a circle is equal to length the measure of the intercepted arc. -Right-angle corollary: If an inscribed angle intercepts a semicircle, the the angle is a right angle. -Arc-intercept corollary: If two insrcibed angles intercep the same arc, then they have the same measure.

__Examples:__ The imaginary arc that is formed from the two points of at the end of the scissors is 16 in. Find the angle: Angle=1/2(16) Angle= 8 degrees

http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html


 * __Chapter 9.4__**

__Objectives:__ -Define angles formed bu secants and tangents of circles. -Develop and use theorems about measures of arcs intercepted by these angles.

__Theorems:__ -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 half the measure of its intercepted arc. -The measure of an angle is formed by tow secants or chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its verticle angle. x1+x2/2 -The measure of an angle formed by two secants that intersect in the exterior of a circle is half the difference the measures of the intercepted arcs.

__Example:__ If there is a circle intercepted by two secants outside of the circle then you would use the formula 1/2(arc1-arc2) to find the interior angle. In this case arc1=10 and arc2=5. Find the measure of the interior angle: 1/2(10-5) 1/2(5) angle=2.5 degrees

http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_CircleSecantTangent.xml


 * __Chapter 9.5__**

__Objectives:__ -Define specieal cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments. -Develop and use theorems about measures of the segmants.

__Theorems:__ -Two segments are tangent to a circle from the same external point, then the segments are equal. -Two secants: w*o=w*o -One secant, one tangent: w*o=t2 -Two Chords: p+1*p+2=p+1-p+2

__Examples:__ The portion of this apple that has been removed to form a mouth is formed by two secants. Two find the measure of the angle of the interior mouth, divide the measure of the arc from the upper lip to the lower lip by two. The arc is 5in. x/2 5/2 angle= 2.5 degrees

http://www.regentsprep.org/Regents/math/geometry/GP14/CircleSegments.htm


 * __Chapter 9.6__**

__Objectives:__ -Develop and use the equation of a circle. -Adjust the equation for a circle to move the center in a coordinate plane.

__Examples:__ -Write an eqaution for a circle with the given center and radius. center: (0,0); radius=6 x+y=r^2 x+y=36 -With the given center:(2,3); radius=4 (x-h)^2+(y-k)^2=r^2 (x-2)^2+(y-3)^2=16

http://library.thinkquest.org/20991/alg2/geo.html