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=Chapter 9=

Definitions Circle- A circle is the set of all points in a plane that are equidstant from a given point in the plane know as the center of the circle.
 * 9.1 Chords and Arcs**

Radius- The radius is a segment from the center of the circle to a point on the circle

Chord- A chord is a segment whose endpoints lay on s circle

Diameter- A diameter is a chord that contains the center of a circle

Arc- An arc is a unbroken part of a circle

Endpoints- Any two distinct points on a circle divide the circle into two arcs

Semi circle- A semi circle is an arc whose endpoints are the endpoints of a diameter

Minor arc- A minor arc of a circle is an arce that is shorter than a semicircle of that circle

Major arc- A major arc of a circle is an arc that is longer than a semi circle of that circle

Central angle- A 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 whose other points in the interior of the angle

Degree measure of arcs- The gegree measure of a minor are 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.

Arc length- If r is the radius of a circle and M is the degree measure of an arc of the circle, then lenght, of a arc is given Arc length formula L=m/360(2pi r) Example Measure is 20 and radius 1 find the length L=20/360(2pi*1) L=2/36(2pi) L=1/18* 2pi/1 L=2pi/18 L= pi/9 L=.35

Example 2 L=15 R=14 (2pi*14) = 87.96 15*360= 5400 5400/87.96= 61.39 M=61.39

Chords and Arcs Theorems- In a circle, or in congruent circles, the arcs of a congruent chords are congruent Secant- Secant to a circle is a line that intersects the circle at two points
 * 9.2 Tangents to Circles**

Tangent- Is a line in the plne of the circle tat intersects the circle at exatly one point, Know as the oint of tangency

Radius and Chord theorem- A radius that is perpendicular to a chord of a circle bisects the chord Example If line BD is 10 and AC bisects it then line BD is split into two segments thats length is 5

Converse of 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

Inscribed angle- This is an angle wose vertex lies on a circle and whose sides are chords of the circle example
 * 9.3 Inscribed Angles and Arcs**

Inscribed Angle Theorem- The measure of an angle inscribed in a circle is equl to half the measure of the intercepter arc

Right angle Corollary- If an inscribed angle intercepts a semi circle, Then the angle is a right angle

Arc intercept Corollar- If two inscribed angles intercept the same arc, then they have the same measure Example-If Arc DE=90 then Angle DAE and DBE = 45 Picture made by jackson loeffel

If a tangent and a secent 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 formed by two secants or chords that intersects in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle X1+X2/2 Example If x1=50 and x2=70 50+70=120/2 So your answer is 60
 * 9.4 Angles Formed by Secants and Tangents**

The Measure of an angle formed by two secants that intersect in the exterior of a circle is half the difference of the measure of the intercepted arcs X1-X2/2

If two segments are tangent to a circle from the same external point, then the segments are equal Line BD and BC are equal
 * 9.5 Segments of Tangents Secants and Chords**

If two secants intersect outside a circle, the product of the lengths of on secant segment and its external equals two secants w*o=w*o Example If segment AC is 10 and EC is 5, BD is 10 what is DE? 10*5=10*x 50=10x 50/10=10x/10 X=5

If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant segment and its external segment equals one secant and 1 tangent w*o=t^2

If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals two chords (pt1)*(pt2)=pt1*pt2 Example AX is 5, XC is 5 BX is 5, XD is unknown 5*5=5*x 25=5x 25/5=5x/5 X=5


 * 9.6 Circles in the Coordinate Plane**

x^2+y^2=36 Center point= (0,0) Radius=6 After you find the radius and center you have to graph it to find the x y-interepts Now you have your four points.

Lets say you need to find the center and radius of a equation like this (x-6)^2+(y-9)^2=100 Now use the standard form of the equation which is (x-h)^2+(y-k)^2=r^2 Your center point should be (6,9) and your radius 10

http://en.wikipedia.org/wiki/Circle Help with Chords http://mathworld.wolfram.com/Chord.html Good example problems http://library.thinkquest.org/20991/textonly/geo/circles.html http://library.thinkquest.org/20991/geo/circles.html
 * Help sites**