hael630

=Chapter 9=

Objectives
1. Define a circle and its associated parts and use them in constructions. 2. Define and use the degree measure of arcs. 3. Define and use the length measure of arcs. 4. Prove a theorem about chords and their intercepted arcs. //If you look at the web page where I found this picture and put your mouse over the circles, it will point out what some of the vocab is.// ~ A minor arc is less then 180° ||  || ~ A major arc is more then 180° ||  ||
 * ===Vocab=== || ===Definition=== || ===Picture/web page=== ||
 * Radius || Is a segment from the center of the circle to a point on the circle.[|click 6] || [[image:bike_wheel_bd.jpg width="93" height="124" link="http://flickr.com/photos/fredandcharlie/137915144/"]] ||
 * Chord || A segment whose endpoints line on a circle. || here is a web page to help you better understand [|click3] ||
 * Diameter || A chord whose endpoints line on a circle. || [[image:diamether.jpg link="http://flickr.com/photos/bitzi/293639007/"]] ||
 * Arc || An unbroken part of a circle. || [[image:cicle_arc.jpg link="http://flickr.com/photos/carlpalmerhull/303449585/"]] ||
 * Endpoints || Any two distinct points on a circle divide the circle into two arcs. The points are called the endpoints ||  ||
 * Semi-circle || An arc whose endpoints are endpoints of a diameter, called a half circle. Named by endpoints and another point that lies on the arc. ||  ||
 * Minor arc || Minor arc of a circle is an arc that is shorter than a semicircle of that circle. A minor arc is named by its endpoints.
 * Major arc || A major arc of a circle is an arc that is longer than a semicircle of that circle. A major arc is named by its endpoints and another point that lies on the arc.
 * 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. || here is a web page to help you better understand [|click 4] ||
 * Intercepted angle || An arc whose endpoints lie on the sides of the angle and whose other points lie in the interior of the angle is the intercepted arc of the central angle. ||  ||
 * Degree measure of arcs || The degree measure of a minor arc is the measure of its central angle. The degree measure of a major arc is 360° minus the degree measure of its minor arc. The degree measure of a semicircle is 180°. ||  ||
 * Circle || 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 || [[image:cicle_sd.jpg link="http://flickr.com/photos/dbarefoot/4764546/"]] ||

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° (2pir) || [|click 11] here is a web page to help you better understand ||
 * ===Definition=== || ===Formula=== || ===Example=== ||
 * Arc Length

Theorems
Chords and Arcs Theorems: In a circle, or in congruent circles, the arcs of congruent chords are congruent.

Example:
Determine the degree measure of an arc with the given length, L, in a circle with the given radius, r. 1. L= 10, r = 50 Steps: 1. 10 = __m__ (2 {pi} 50) 360 2. divide everything by (2 {pi} 50), which crosses itself out 3. 360(1÷20{pi}) = (m ÷ 360) 360 360 and 360 cross each other out 4. = 226.19

Links
Here is a link to help you understand better[| click].

Objectives
1. Define tangents and secants of circles. 2. Understand the relationship between tangents and certain radii of circles. 3. Understand the geometry of a radius perpendicular to a chord of a circle.


 * ===Vocab=== || ===Definition=== || ===Picture/ web page=== ||
 * Secant || A secant to a circle is a line that intersects the circle at two points. || here is a web page to help you better understand [|click 6] ||
 * Tangent || A tangent is a line in the plane of the circle that intersects the circle at one point. || here is a web page to help you better understand [|click 7] ||
 * Point of tangency || Is the point where the tangent line intersects the circle in only one spot ||  ||

Theorems
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 chords of a circle bisects the chord. Converse of 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.

Example:
picture made by me segment ry is perpendicular to xz at w, and r is the center of the circle if RY= 8 andRW = 3 what is xw? xw = 5 because 8-3 = 5

Chapter 9.3
//if you look on the web page were I found this picture and put your mouse over the picture, it will point out some vocab.//

Objectives
1. Define inscribed angle and intercepted arc. 2. Develop and use the inscribed angle theorem and its corollaries.
 * ===Vocab=== || ===Definition=== || ===picture / web page=== ||
 * Inscribed angle || A angle whose vertex lies on a circle and whose sides are chords of the circle. || here is a web page to help you better understand [|click 8] ||

picture made by me If the major arc is 240, what is angle a? Steps: __240__ = 120 2 angle a is 120°

Theorems
Inscribed Angle Theorem: The measure of an angle inscribed in a cicle is equal to the measure of the intercepted arc.

Corollary
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.

Objectives
1. Define angles formed by secants and tangents of circles. 2. Develop and use theorems about measures of arcs intercepted by these angles. 2 ||  || 2 || picture made by me If arc 1 is 180 and arc 2 is 70 what is angle a and b? 70+180 = 125 2 angle a and b are both 125 because of vertical angles. || 2 ||  ||
 * ===Case=== || ===Formula=== || ===Picture/example=== ||
 * 1 vertex on the circle || __1__
 * 2 vertex inside the circle || __x1+x2__
 * 3 vertex on the outside of the circle || __x1-x2__

Theorems
Vertex on circle-secant and tangent case one: 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 the measure of its intercepted arc.

Vertex inside circle-two secants case two: The measure of an angle formed by two secants of chords that intersect in the interior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle.

Vertex outside circle-two secants case 3: The measure of an angle formed by two secants that intersect in the exterior of a circle is half the of the measures of the intercepted.

1. The measure of a secant-tangent angle with its vertex outside the circle is __x1+x2__

2. The measure of a tangent-tangent angle with its vertex outside the circle is __x1-x2__

Objectives
1. Define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments. 2. Develop and use theorems about measures of the segments.

Theorems
1. If two segments are tangent to a circle from the same external point, then the segments are equal. 2. If two secants intersect outside a circle, the product of the length of one secant segment and its external segment equals the product of the length of one secant segment and its external equals the product. ( 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 equals the product of the length of the secant segment and its external segment quals the product. ( Whole x Outside = Tangent Squared ) 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 one chord. (ab)=(cd)

Example:
What is x?

Steps: 2(x)=(4)(8) 2x=32 32 ÷ 2 x = 16

picture made by me

Objectives
1. Develop and use the equation of a circle. 2. Adjust the equation for a circle to move the center in a coordinate plane.

Example:
1. Equation for a circle with a center at the origin is, x² + y² = r². r being the radius find the x and y intercepts intercepts: (2,0), (-2,0) , (0,2) , (0,-2) equation: x² + y² = 2² pictures made by me



2. Equation for a circle with theh center not at the origin but at a new point (h,k) the center, (x-h)² + (y-k)² = r², r being the radius find the x and y intercepts intercepts: There are no x and y intercepts equation: (x-5)² + (x-3)² = 1.25