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__Chapter 9 Wiki__

__9.1 Chords and Arcs__ __Objectives-__ Define a circle and its associated parts, then use them in constructions. Get to know more about the degree measure of arcs. Learn more about the is of the length measure of arcs. Prove a theorem about chords and their intercepted arcs.

__Circle__- A circle is the set of all points that are equidistant from a given point in the plane known as the center of the circle. The radius is the segment from the center of the circle to a point on the circle.

__Chord__- Segment whose endpoints line on a circle.

__Diameter-__ Chord that contains the center of a circle.

__Central Angle-__ Is an angle in the plane of a circle whose vertex is the center of the circle.

__Intercepted Arc-__ When the arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle.

__Arc-__ The unbroken part of the circle.

__Endpoints-__ Any 2 distinct points on a circle divide the circle into 2 arcs.

__Semicircle-__ Arc whose endpoints are endpoints of a diameter. Usually called a half circle because of its endpoints and another point that lies on the arc. __Minor Arc-__ Arc that is shorter than a semicircle of that circle. Just like the semicircle the minor arc is named by its endpoints.

__Major Arc-__ An arc that is longer than a semicircle of the circle. Named by its endpoints and another point that lies on the arc.

__Degree Measure Of Arcs-__ The degree measure of a minor arc is the measure of its central angle. Major arc is 360 degrees take away the degree of its minor arc. The semicircles degree measure will always be 180 degrees.

__Arc Length-__ L= M/360 degrees (2*3.14*r) L= Length r=Radius M=Degree measure of the arc.

__Chords and Arcs Theorem-__ In congruent circles or just circles, the arc of congruent chords are congruent.

__Converse Of Chords and Arcs Theorem-__ The chords of congruent arcs are 1/2. http://www.flickr.com/photos/murky/5040802/ The above link is to a real example of an arc.

Example: These two measures were found from their central angles.



M AC=130 M BC=70 To find the measure of the circle of M ABC add the two measures together. So 130=70=200 <=== answer.

__9.2 Tangents To Circles__ __Objectives-__ Define tangents and secants of circles. Get to know the relationship between tangents and certain radii of circles. Understand the geometry of a radius that is perpendicular to a chord of a circle.

__Secant__- Is a line that intersects the circle at two points.

__Tangent-__ Is a line in the plane of the circle which intersects the circle at exactly one point. This point is known as the point of tangency.

__Tangent Theorem__- If a line is in the plane of the circle which intersects at exactly one point, then the line is congruent to a radius of a circle drawn to the point of tangency.

__Radius & Chord Theorem-__ A radius that is perpendicular to a chord of a circle bisects the chord.

__Converse Of The Tangent Theorem-__ If a line is perpendicular to a radius of a circle at its endpoint on the circle at its endpoint on the circle, then the line is tangent to the circle.

__Theorem__- The perpendicular bisector of a chord passes through the center of the circle

In this diagram CP is the radius and there is no chord.

__9.3 Inscribed Angles and Arcs__

__Objectives-__ Define inscribed angle and intercepted arc. Develop and use the inscribed angle theorem and its corollaries.

__Inscribed Angle-__ Is an angle in whose vertex is on the circle and whose sides are chords of the circle.

__Inscribed Angle Theorem-__ The measure of an angle inscribed in a circle is equal to 1/2 the measure of the intercepted arc.

__Right Angle Corollary-__ If the inscribed angle intercepts a semicircle the angle will be right.

__Arc-Intercept Corollary-__ If 2 inscribed angles intercept the same arc, they will have the same measure.

Example. Measure VXY is inscribed in circle Q and intercepts VY. By the inscribed angle theorem. mVXY=1/2mVY=1/2(45%)=22 1/2%.

__9.4 Angles Formed by Secants and Tangents__ __Objectives-__
 * Define angles formed by secants and tangents of circles.
 * Make and use theorems about measures of arcs intercepted by these angles.

__Theorem Page 589__ If a tangent and a secant intersect on the point on a circle at the point of tangency, then the measure of the angle formed is 1/2 the measure of the intercepted arc.

__Theorem 9.4.2 Page 590__ The measure of an angle formed by two secants or chords that intersect in the interior of a circle is 1/ the sum of the measures of the arcs intercepted by the angle and its vertical angle.

__Theorem 9.4.3 Page 590__ The measure of an angle formed by 2 secants that intersect in the exterior of a circle is 1/2 the difference of the measures of the intercepted arcs.

__9.5 Segments of Tangents, Secants, and Chord__
 * __Tangent__ __Segment-__ Is a segment that touches the circle once.
 * __Secant Segment-__ Intercepts with the the tangent segment and also touches the circle twice.
 * __External Secant Segment-__ The part of the secant segment that is outside of the circle.
 * __Chord-__ Interior part of the secant segment that touches the circle twice.

__Theorem 9.5.1 Page 601__ If 2 segments are tangent to a circle from the same external point, then the segments are of equal length.

__Theorem 9.5.2 Page 601__ If 2 secants intersect outside of a circle, the 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.

__Theorem 9.5.3 Page 602__ If a secant and a tangent intersect outside of a circle, then the product of the lengths of the secant segment and its external segment equals the length of the tangent segment squared.

__Theorem 9.5.4 Page 602__ If 2 chords intersect inside a circle, then the product of the lengths of the segments of the chord equals the product of the lengths of the segments of the other chord.

__9.6 Circles In The Coordinate Plane

__<======Numbers are changed to 6 not 5!!!

__Now to Solve this Equation you would use....__

1. x2+y2=36. Now you are going to know that the x and y are the center of the graph and also you now know the center of the graph is (0,0). Then you would solve for x. x2+o2=36 x2=36 Take the square root of 36 and you get................................x=+-5