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CHAPTER 9 SECTION 9.1 Chords and Arcs Objectives

Example1:
 * define and associete parts and there constructions.
 * define the degree and measure fo arcs.
 * define measure the length measure of arcs.
 * prove therom chods and intercepted arcs.
 * Circle: set of points in a plane equildistant from a given point.This plane is known as a circle.
 * Radius: (Plural Radii) segment form center of circle.
 * Chord : segment with endpoints on a circle.
 * Diameter: chord contains the center of a circle.
 * Arc: unbroken part of circle.
 * Endpoint: starting point of a ray.
 * Semi-circle: The arc of a circle whose endpoints are the end points of a diameter.
 * Minor arc: An arc of a circle that is hsorter than a semicercle of that circle.
 * Major arc: An ac of a circle that is longer than a seicircle of that circle.
 * Central angle: Of a circle is an angle in the plane of a circle whose vertex is the center of the circle.
 * Intercepted arc: An arc whose endpoints lie on the sides of the angle and whoses other points lie in the interior of the angle is the intercepeted arc ot the central angle
 * Degree measure of arcs: The degree measre of a minor arc is the measure of its central angle. Te defree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180

mAT= 50° mDC=30° mRTS=mRT+mTS=50°+30°=80° http://library.thinkquest.org/10030/13arcsandc.htm http://www.mathematicshelpcentral.com/lecture_notes/college_geometry_folder/circles.htm http://library.thinkquest.org/C006354/9_2.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html > > > > Section 9.2 > Tangents to Circles > Objectives > Define tangents and Secants. > Understand the relationships and radii of circles. > Understand radius perpendicular to chord and circle. Example: (AX)²+4²=20² (AX)²=20²-4² (AX)²=384 AX=196 http://library.thinkquest.org/C006354/9_2.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm http://library.thinkquest.org/10030/13tangen.htm
 * Secant: to a circle is the line that intersects the circle at two points.
 * Tangent: is a line in the plane of the circle that intersect 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 perpendicular to a radius of the circle drawn to the point of tangency.
 * Radius and Cord Theorem: A radius that is perpendicular ot a cord of a circle intercest the chord.
 * Convverse of Tangent Theorem: If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

Section 9.3 Inscribed Angles and Arcs Objectives Example: M‹XYZ=½mÂÊ=½(45°)=22½ http://library.thinkquest.org/C006354/9_2.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm http://www.geom.uiuc.edu/~dwiggins/conj44.html http://www.ies.co.jp/math/java/geo/enshukaku/enshukaku.html
 * Define inscribed angle and intercepted arc.
 * Develop the Inscribed Angel Theorem and corollaries.
 * Inscribed Angle: Is an angle whose vertex lies on a circle and whose sides are chords of the circle.
 * Inscribed Angle Theorem: The measure of an angle inscribed in a circle is equal to ½ the measure of the intercepeted arc.
 * Right-Angle Corollary: If an inscribed angle intercepts a semicircle, then the angle is a right angle.
 * Arc-Intercept Corollary: If two incrbed angles intercept te same arc, then they have the same measure.

Section 9.4 Angles Formed by Secants and Tangents Objctives Example1: Secant and a tangent Vertex is on the circle
 * Define angles by secants and tangents
 * Develop theorems about measure of arcs and interceped angle
 * Theorem 9.4.2: 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 ½ the measure of its intercepted arc.
 * Theorem 9.4.2: The measure of an angle formed by two secants or chords that intersects in the interior of a circle is bland and blank of hte measures of the arcs interceped by the angle and its vertical angle.

Two Secants Vertex on the circle



Example 2: Two Secants Vertex on the inside of the circle



Example 3: Two Tangents Vertex on outside of circle

Two Secants Vertex on outside

Secant and Tangent Vertex outside circle

http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html http://library.thinkquest.org/C006354/9_2.html http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm http://www.algebralab.org/lessons/lesson.aspx?file=Geometry_CircleSecantTangent.xml http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php

Section 9.5 Segments of Tangents, Secants, and Chords Objectives
 * Define special cases of segments, like Secant-secant, Secant-tangent, and Chord-chord segments
 * Develope theorems for measures of the segment
 * Theorem 9.5.1: If two segments are tangent to a circle form the same external point, then the segments intersect.
 * Theorem 9.5.2: If two secants intersect outside a cirlce, the products of the lenghs of one secant segment and its external segment equals the whole.
 * Theorem 9.5.3: If asecant and a tangent intersect outside a circle, then the product of the lenghts of the secant segment and its external equals the tangent squared.
 * Theorem 9.5.4: If two chords intersect one of its endpoints on the nside a cirlce, then the product of the lenghs of the segments of one chord equals the angles.
 * Tangent segment: A segment that is contained by a line tangent to a cirlce and has one of its endpoints on the circle.
 * Secant segment: A segment that contains a chord of a cirlce and as one endpoint exterior to the circle and the other endpoint 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.

Example: EX×GX=FX×HX 1.31×0.45=1.46×HX 1.46×HX=1.46×HX 1.46×HX=0.5895 HX=0.40 http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html http://library.thinkquest.org/C006354/9_2.html http://www.regentsprep.org/Regents/math/geometry/GP14/CircleSegments.htm http://www.regentsprep.org/Regents/math/geometry/GP14/PracCircleSegments.htm take the quiz

Section 9.6 Circles in the Coordinate Plane Objectives: Examples: Equation fokr a circle with a cener at origin X²+Y²=R² C=(0,0) R=? X²+Y²=25 R=5
 * Develop the equation of a circle.
 * Adjust equation to move center in a coordinate plane.

Example 2: Equation of a circle with a center not on the origin (X-H)²+(Y-K)²=R² (H,K) possitive R=3² (X-7)²+(Y-3)²=3² (H,K)=(7,3) http://library.thinkquest.org/C006354/9_2.html http://jwilson.coe.uga.edu/EMT668/EMT668.Student.Folders/RothJennifer/TenDayUnit/Unit.html http://regentsprep.org/regents/mathb/5A1/CircleAngles.htm http://web.nafcs.k12.in.us/users/NAHS/bbanet/Sec11-5.ppt http://library.thinkquest.org/20991/alg2/geo.html#circles