MEEM115

=CHAPTER 9 Wiki= By: Emily Mechelke

__9.1 Chords and Arcs__ [[image:brick_circle.jpg width="277" height="183" link="http://www.flickr.com/photos/alidasphotos/486391421/"]]
= = Circles: A formal Definition A circle is the set of all points in a plane that are equidistant from a given point in the plane known as the center of the circle. Definitions: centeral Angle and Intercepted Arc Definitions: Degree Measure of Arcs The degree measure of a minor arc is the measure of its central angle.
 * Radius:** Is a segment from the center of the circle to a point on the circle.
 * Chord:** is a segment whose endpoints line on a circle.
 * Diameter**: is a chord that contains the center of a 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 whose other points lie in the interior of the angle is the intercept.
 * 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.
 * Chords and Arcs theorem:** In a circle or in congruent circles the arcs 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.

Click [|here] to get a picture to all the words abo ve in this section!!

__9.2 Tangents to Circles__
Click [|here] for sweet picture!
 * Secant:** to a circle is a line that intersects the circle at two points.
 * Tangent:** is a line in the plane of the circle that intersects 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 Chord Theorem:** A radius is perpendicular to a chord of a circle.
 * Converse of the Tangent Theorem:** If line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is perpendicular to the circle.
 * Theorem:** The perpendicular bisector of a chord passes through the center of the circle.

__**9.3 Inscribed Angles and Arcs**__
An inscribed angle is an angle whose vertex lies on a circle and whose sides are chords of the circle. Click [|here] to get picture of inscribed angles and Arcs
 * Inscribed Angle Theorem:** The measure of an angle inscribed in a circle is equal to the measure of the intercepted arc.
 * Right-Angle Corollary:** If an inscribed angles intercept 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.

**__9.4 Angles Formed by Secants and Tangents__**
If a tangent and a secant 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 9.4.2**__ The measure of an angle formed by two secants or chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle. __**Theorem 9.4.3**__ The measure of an angle formed by two secants that intersect in the exterior of a circle is half the difference of the measures of the intercepted arcs. __**Theorem 9.4.4**__ The measure of a secant- tangent angle with its vertex outside the circle is half the difference of the measures of the intercepted arcs. __**Theorem 9.4.5**__ The measures of a tangent-tangent angle with its vertex outside the circle is half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180°. click[|here] for more help and better pictured examples.
 * __Theorem:__** If a tangent and a secant intersect on a circle at the point of tangency, then the measure of the angle formed is the measure of its intercepted arc.
 * __Theorem 9.4.1__**

__**9.5 Segments of Tangents, Secants and Chords**__
Tangent line to the cemetary. If two segments are tangent to a circle from the same external point, then the segments are equal length. __**Theorem 9.5.2**__ If two secants intersect outside a circle, then the product of the lengths of one secant segment and external segment equals the product of the lengths of the other secant segment and its external segment. __**Theorem 9.5.3**__ 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 the length of the tangent segment squared. __**Theorem 9.5.4**__ If two chords intersect inside a circle, then the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord. click[|here] and check out some cool examples to better your learning! =__9.6 Circles in the Coordinate Plane:__=
 * __Theorem 9.5.1__**

Create an equation to graph a circle on the origin.

 * Create an equation to graph a circle not on the origin.**

__Equations& Examples:__
When the center of the circle is at the origin (0,0) X² + Y² = r ² When the center of the circle is not at the origin: Examples: (X - h)² + (Y - k)² = r ² (h,k) = the orgin