SE+GR+5+22

=CHAPTER 9.1(CHORDS AND ARCS)= OBJECTIVES- Define a circle and its associated parts, and use them in constructions. OBJECTIVES- Defined and use the degree measure of arcs. OBJECTIVES- Define and use the length measure of arcs. OBJECTIVES- Prove a theorem about chords and dtheir intercepted arcs. //**VOCAB**//

ARC- An unbroken part of a circle. (Any two distinct points on a circle divide the circle into two arcs.) MINIOR ARC- An arc that is shorter than a semicircle of that circle. (named by its endpoints.) MAJOR ARC- An arc that is longer than a semicircle of that circle. (named by its endpoints and another point that liew on the arc.) ARC LENGTH- The measure of the arc of a circle in terms of linear units, such as centimeters ARC MEASURE- The measure of an arc in a circle in terms of degrees. CHORD- A segment whose endpoints lie on a circle. INTERCEPTED ARC- An inscribed circle in a triangle is inside the triangle and touches each side at one point; a circle is inscribed in a polygon if eacho side of the polygon is tangent to the circle.

__DEFINITION OF A CIRCLE__ A circle is the set of all points in a plane that are ewuidistant 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 circleto a point on the circle. A CIRCLE is a segment whos endpoints line on a circle A DIAMETER is a chord that contains the center of a circle.



__DEFINITIONS OF CENTRAL ANGLES AND INTERCEPTED ARCS__ A CENRAL ANGLE of a circle is an angle in the plane of a circle whose vertex is the center of the circle. And arc whose endpoints lie in the interior of the angle is the INTERCEPTED ARC f the central angle.

__DEFINITION OF DEGREE MEASURE OF ARCS__ The degree measure of a measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180°

__THE CHORDS AND ARCS THEOREM__ In a circle, or in congruent circles, the arc of congruent chords are congruent.

__THE CONVERSE OF THE CHORDS AND ARCS THEOREM__ In a circle or in congruent circles. the chords of congruent arcs are congruent.

__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 given by the following: L= M÷360° (2×Pi×r)
 * formula**

L= 40 ÷ 360 (2 x Pi x 50) L= 34.91feet in lengh before you sink in quicksand
 * EXAMPLE-** You need to figure out long the lenth of grass is around the lake. If you walk to far you will sink into the quicksand! How long does the grass stretch?

=CHAPTER 9.2 (TANGENTS TO CIRCLES)= OBJECTIVES- Define //tangents// and //secants// if a circle. OBJECTIVES- Understand the relationship between tangents and certan radii of circles. OBJECTIVES- Understand the geometry of a radius perpendicular to a chord of a circle.

//**VOCAB**//

SECANT- A line that intersect the circle at two points TANGENT- A line in a plane of the circle that intersects the circle at exactally one point. POINT OF TANGENCY- The exact point where the tangent intersects the circle. CENTER OF A CIRCLE- The point inside a circle that is equidistant from all points on the circle. CENTRAL ANGLE OF A CIRCLE- An angle formed by two rays originating from the center of a circle.

__SECANTS AND TANGENTS__ A SECANT to a circle is a line that intercets the circle at two points. A TANGENT is a line in the plane of the circle that intersects the circle at exactally one point, which is known as the POINT OF TANGENCY.

__THE 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.

__THE RADIUS AND CHORD THEOREM__ A radius that is perpendicular to a chord of a circle bisects the chord.



__THE CONVERSE OF THE TANGENT THEOREM__ If a line is perpendicular to a radius of a circle at its endpoints on the circle, then the line is tangent to the circle.

__THEOREM__ The perpendicular bisector of a chord passes throught the center of the circle. = = = =

=CHAPTER 9.3(INSCRIBED ANGLES AND ARCS)=

OBJECTIVES- Define //inscribed angle// and //intercepted arc.// OBJECTIVES- Develop and use the Inscribed Angle Theorem and its corollaries.

//**VOCAB**//

INSCRIBED ANGLE- An angle whose vertex lies on a circle and whos sides are chords of the circle. SECAN SEGMENT- A semgnet that contains a chord of a circle and has one endpoint exterior to the circle and the other endpoint on the circle. ENDPOINTS- The points of the arc.

__THE INSCRIBED ANGLE THEOREM__ The measure of and angle inscribed in a circle is equal to one-half the measure of the intercepted arc.

__RIGHT ANGLE COROLLARY__ If and inscribed angle intercepts a semicircle, then the angle is a right angle.

__ARC-INTERCEPT COROLLARY__ If two inscribed angles intercept the same arc, then they have the same measure.

=CHAPTER 9.4(ANGLES FORMED BY SECANTS AND TANGENTS)=

OBJECTIVES- Define angles formed by secants and tangents of circles. OBJECTIVES- Develop and use theorems about measures of arcs intercepted by these angles.

//**VOCAB**//

RADIUS- A segment that connects the center of a circle with a point on the circle; one-half the diameter of a circle. TANGENT SEGMENT- A segment that is contained by a line thangent to a circle and has one of its endpoints on the circle.

__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 one-half 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 one-half the 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 the exterior of a circle is one-half the of the measures of the intercepted arcs.

__THEOREM__ The measure of a secant-tangent angle with is v ertex outside the circle is one-half the difference of the measures of the intercepted arcs.

__THEROEM__ The measure of a tangent-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180°

=CHAPTER 9.5(SEGMENTS OF TANGENTS, SECANTS AND CHORDS)= OBJECTIVES- Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments. OBJECTIVES-Develop and use theorems about measures of the segments.


 * //VOCAB//**

DIAMETER- A chord that passes through the center of a circle; twice the length of the radius of the circle. EXTERNAL SECANT SEGMENT- The portion of a secant segment taht lies outside the circle.



__THEOREM__ If two segments are tangent to a circle from the same external point, then the segments are of equal length.

__THEOREM__ If two secants intersect outside a circle, then the product of the lengths of one secant segment and its external segment equals the prodcut of the lenghtsof the other secant segment and its external segment. (Whole x Outside= Whole x Outside)

__THEOREM__ If a secant and a tangent intersect outside a circle, then the product of the lenghts of the secant segment and its external segments equals the lengh of the tangent segment squared. (Whole x Outside= Tangent Squared)

__THEOREM__ If two chords intersect inside a circle, then the product of the lenghts of the segments of one chord equals the product of the lenghts of the segments of the other chord.



=CHAPTER 9.6(CIRCLES IN THE COORDINATE PLANE)= OBJECTIVES- Develop and use the equation of a circle. OBJECTIVES- Adjust the equation for a circle to move the center in a coordinate plane.

//**VOCAB**//

CIRCLE- Te set of points in a plane taht are equidistant from a given point known as the center of the circle. SEMICIRCLE- And arc whos endpoints are endpoints of a diameter. (A semicircle is informally called a half-circle.)

http://www.geom.uiuc.edu/%7Edwiggins/conj42.html good link for tangent conjectures http://www.math.com/tables/geometry/circles.htm to understand circles better http://library.thinkquest.org/20991/geo/circles.html segments, tangents, secants and MORE!!!!!!!!!!!!!!!!! http://www.sparknotes.com/math/geometry2/theorems/section4.rhtml geometric theorems