rakr317

=Chapter 9=

__9.1 - Chords and Arcs__
//Objectives;// >Define a circle and its associated parts, and use them in constructions. >Define and use the degree measure of arcs. >Define and use the length measure of arcs. >Prove a theorm about chords and their intercepted arcs.

//Definitions;// >Circle : the set of all points in a plane that re equidistant from a given point in the plane known as the center of the circle. >Radius : 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 : 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 circle and whose other points lie in the interior of the angle. >Minor Arc : an arc that is shorter than a semicircle of that circle. It's named by its endppoints. >Major Arc : an arc that is longer than a semicircle of that sircle. It's named by its endpoints and another point that lies on the arc. >Semicircle : an arc whose endpoints are endpoints of a diameter. a semicircle is informally called a half-circle. >Degree Measure of Arcs : the degree measure of a minor arc is the measure of its central anle. 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 the length, //L//, of the arc is given by the following : //L// = //M///360 (2(pi)//r//)

**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 are congruent.

__**EXAMPLE.**__
A circle has an arc, //RT//, which equals 90 degrees and another arc, //ST//, which equals 100 degrees. What does //RS// equal? 360º - 90º - 100º = 170º So, //RS =// 170º [|website.]

__9.2 - Tangents to Circles__
//Objectives;// >Define tangents and secants of circles. >Understand the relationship between tangents and certain radii of circles. >Understand the geometry of a radius perpendicular to a chord of a circle.

//Definitions;// >Secant : a line that intersects the circle at two points. >Tangent : 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 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, then the line is tangent to the circle.

**Theorem**
The perpendicular bisector of a chord passes through the center of the circle.

__**EXAMPLE.**__
//BC// is tangent to circle //A// at //B//. If //BA// is 2 and //CA// is 3, find //KL.//

2^2 + B^2 = 3^2 4 + B^2 = 9 B^2 = 5 ≈2.24 = = [|website.]

[[image:6050999_fabecc7a16.jpg width="225" height="178" link="http://www.flickr.com/photos/45688285@N00/6050999/"]]
//Objectives;// >Define //inscribed angle// and //intercepted arc.// >Develop and use the Inscribed Angle Theorem and its corollaries.

//Definintions;// >Inscribed Angle : 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 one-half the measure of the intercepted arc.

**Right-Angle Corollary**
If an 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.

__**EXAMPLE.**__
Arc //AC// = 50º and Define angles formed by secants and tangents of circles. >Develop and use theorems about measures of arcs intercepted by these angles.

//Theorems;// 1. If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle is one-half the measure of its intercepted arc. 2. The measure of an angle formed by two secants or chords that intersect in the interior of a circle is one-half the sum of the measures of the arcs intercepted by the angle and its vertical angle. 3. The measure of an angle formed by two secants that intersect in the exterior of a circle is one-half the difference of the measures of the intercepted arcs.

__**EXAMPLE.**__
Arc //AB// equals 60º and arc //CD// is 40º. Find the measure of <//AVB//. 60º + 40º equals 100º/2 50º So, m<//AVB// is 50º along with <//CVD// because of the vertical angles theorm.

[|website.]

__9.5 - Segments of Tangents, Secants, and Chords__


//Theorems;// 1. If two segments are tangent to a circle from the same external point, then the segments are congruent. 2. If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment is **whole**x**outside**=**whole**x**outside** 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 is **whole**x**outside=t^2** 4. If two chords intersect inside a circle, then the product of the lengths of one chord is **A**x**B**=**C**x**D

__EXAMPLE.__

** Solve for x. 4x3 = 5x 12 equals 5x X=2.4

[|website.]

__9.6 - Circles In The Coordinate Plane__
x^2 + y^2 = r^2 x = x value y = y value coordinate pair (x,y) r = radius

//Write an equation for the circle with the given center and radius.// (0,0) r equals 6 x^2 + y^2 = 36

//Find the center and radius of each circle.// (x+2)^2 + (y-3)^2 = 100 (-2,3) radius is 10

(x-2)^2 + (y-3)^2 = 16
 * Graph the following equation.**

[|website.]


 * All non-linked photos are made by me in paint :)