lode35

=Chapter 9=

9.1Chords and Arcs

 * Circle-** It is the set of all points in a plane that are equidstant.
 * Radius-** Is a segment from the center of the circle to a point on the circle.
 * Chord-** Is a segemnt whose end points line oa circle.
 * Diameter-** Is a chord that contines the center of a circle.
 * Arc-** Is an unbroken part of a circle.
 * Endpoints-** Any two distinct points on a circle divide the circle into two arcs.
 * Semi-circle-** Is an arc whose endpoints are enpoints of a diameter.
 * Minor Arc-** A circle is an arc that is shorter than a semicircle of that circle.
 * Major Arc-** A circle is an arc that is larger then a semicircle of that circle.
 * Central Angle-** A circle is an angle in th plain of a circle whose vertex is the center of the circle.
 * Intercepted Arc-** Central angles.
 * Degree Measure Of Arcs-** The degree of a minor arc is the measure of its central angle, the degree measure of a major arc is 360 degrees minus the degree measure of its minor arc. The degree measure of a semicircle is 180 degrees.


 * Example 1- Find the measure of RT, TS, RTS.**
 * Solution: The measure of RT and TS are found

Example 2** The lenght of the arc is 1/20 of the circumfrence of the circle. Remeber that C=2Pir Lenght of arc=1/20(2Pi×170) =17PiEqivlent53.4Eqivlent53mn

L=M÷360(2Pir) Chords and Arcs Theorems- In a circle or in congruent circles, the arcs of congruents chords are
 * Arc Length-** If r is the radius of a circle and M is the degree measure of an arc the circle, then the length, L, of the arc is given by the following.

**9.2 Tangents and Circles**

 * Secant-** A circle is a line that intersects the circle at two points.
 * Tangent-** Aline in the plane of the circle that intersects the circle at the exactly one point.
 * Point Of Tangency-** It is a tangent.
 * Tangent Theorem-** If a line is tangent to a circle, then the line is perpendicular to a radius of the circle drown to the point of tangency.

(AX)²+3²=5² (AX)²=5²-3² (AX)²=16 AX=4
 * Example 1- Circle P has a radius of 5 in. and PX is 3 in.PR is perpendicular to AB at point X. Find AB.**
 * Solution:**


 * Convverse of Tangent theorem-** If a line is perpendicular to a radius pf a circle at its endpoints on the circle, then the line is tangent to the circle.

[[image:http://farm1.static.flickr.com/188/364933756_2c61fca796.jpg?v=0 width="344" height="279" link="http://www.flickr.com/photos/35636897@N00/364933756/"]]
Find the measure of <XVY. <XVY is inscribed in P and intercepts arc XY. m<XVy (1/2)m(arc)XY (1/2)(45) = 221/2
 * 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 half the measure of the intercepted arc.
 * Example 1:**

Right-angle Corollary-** If two inscribe angles intercept a semi-circle, then the angle is a right angle.
 * [[image:angelawrisky1.JPG]]
 * Arc-Intercept Corollary-** if two inscribed angles intercept the same arc, then they have the same measure.

9.4 Angles Formed By Secants and Tangents
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 1/2 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 1/2 the sum of the measures of the arcs 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 1/2 the different of the measure of the intercepted arcs.
 * Theorem 9.4.1-**
 * Theorem 9.4.2-**
 * Theorem 9.4.3-**

Example:
Find angle XYZ in each figure. Angle XYZ is formed by a secant and a tangent that intersect on the circle. measure of angle XYZ is ½ measure of arc XY (200°) = 100° Angle XYZ is formed by two secants that intersect inside the circle. measure of angle XYZ is ½ (measure of arc XY + measure of arc PQ)1/2 (100° + 20°) = 60° Angle XYZ is formed by two secants that intersect outside the circle. measure of angle XYZ is ½ (measure of arc XZ - measure of arc PQ)1/2 (100° - 50°) =25°
 * Circle A-**
 * Circle B-**
 * Circle C-**

9.5 Segments Of Tangents, Secants, and Chords

 * Tangent Segement-** A segment that is contained by a line tangent ot a circle and has one of its endpoints on the circle.
 * Secant Segement-** A segment that containes a chord of a circle and has one endpoint exterior ot the circle and the other endpoint on the circle.
 * External Secant Segment-** The portion of a cecant segment that lies outside the circle.
 * Chord-** A segment whose endpoints lie on a circle.

**Example:**
Theorem 9.5.1 If two segment are tagent to a circle from the same external point, then the segments are of equal length. If two secants intersect outside a circle, the the product of the lengths of one secant segment and its external segment equal the product of the length of the other secant segment and its external segment(w×o=w×o). If a secant and a tangem]nt intersect outside a circle, then the product of the lengths of the secant segment and its external segments segemnt squred(w×o=t²).
 * Theorem 9.5.2**
 * Teorem 9.5.3**

Example:
Given: x²+y²=81 To find the center and radius of this circle we have to find the square root of 81. Which is 9 and that is our radius. Since the center is at (0,0) (i found that out because x and y dont have any numbers before them). Next you graph it. Your center is at (0,0) and your radius is 9. So on the x and y axsis extend out.