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=CHAPTER 9=


 * __Chapter 9.1__**

//__Objectives:__//
- Define a circle and its associated parts and use them in constructions. - Define and use the degree measures of arcs. - Define and ust the length measure of arcs. - Prove a theorem about chods and their intercepted arcs.

__//Words To Know://__
- Circle: Set of points in a plane that are equidistant from a gven point known as the center of the circle. - Radius: A segment that connects the center of a circle with a point on the circle; one-half of the diamter. - Chord: A segment whose end points lie on a circle. - Diameter: A chord that passes thorugh the center of a cicle and is twice the length of the radius. - Arc: An unbroken part of a circle. - Endpoints: A point at an end of a segment or the starting point of a ray. - Semi-circle: The arc of a circle whose endpoints are the endpoints of a diameter. - Minor Arc: An arc of a circle that is shorter than a semicircle of that circle. - Major Arc: An arc of a circle that is longer than a semicircle of that circle. - Central Angle: An angle formed by two rays originating from the center of a circle. - Intercepted Arc: An arc whose endpoints lie on the sides of an inscribed angle. - Degree Measure of Arcs: The measure 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 central angle.

//__Examples:__//
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 * __Chapter 9.2__**

//__Objectives:__//
- Define tangents and seceants of circles. - Understand the relationship between tangents and certain readii of circles. - Understand the geometry of a radius perpendicular to a chord of a circle.

//__Words To Know:__//
- Secant: A line that intersects a circle at two points - Tangent: In a right triangle, the ratio of the length of the side oposite an acute angle to the length of the side adjacent - Point of Tangency: The point of intersection of a circle or sphere witha tangent lineor plane - Tangent Theorem: If a line is tangent to a circle, then the lineis 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 Tangent Theorem: If a line is perpendicular to a radius of a circle at its end point on the circle, then the line is tangent to the circle

__**Chapter 9.3**__

//__Main Objectives:__//
- Define inscribed angles and arcs. - Understand and use the inscribed angles theorem.

__//Words To Know://__
- Inscribed Angle: Is an angle whose vertex lies on a circle and whose sides are chords of the circle - Inscribed Angle Theorem: The inscribed is measured half as much as the intercepted arc. - Right-Angle Corollary: When a inscribed angle is inside a semi-circle the angle is a 90°(right angle). - Arc-Intercept Corollary: When two inscribed angles are intercepting the same arc, it makes them the same measure. Arc BC is 78. Find the measure of angle BAC.

__**Chapter 9.4**__

//Objectives://
- Define angles fromed by tangents and secants. - Develop and use some theorems about measures of arcs intercepted by these angles.

//__Theorems:__//
- When two lines intersect __outside__ the circle the measure of the intercepted angle is half the difference of the measure of the arcs intercepted by the angle and its vertical angle[ (X1°-X2°)/2. - When two lines intersect __on__ the circle the measure of the intercepted angle is half the measure of the intercepted arc (X°/2). - When two lines intersect __inside__ the circle the measure of the intercepted angle is half the sum of the measure of the arcs intercepted by the angle and its vertical angle[ (X1°+X2°)/2.


 * __Chapter 9.5__**

//Objectives://
- Define special cases of segments related to circles. - Develop and use theorems about the cases of segments related to circles.

//Theorems://
- If there are two tangents from the same external point, then the segments have an equal length. - If two secants intersected outside a circle, then the product of the whole segment and the outside segment equals the product of the whole segment and the outside segment.(Whole × outside = Whole × outside) - If two chords intersect inside the circle, then the product of the whole segment and the outside segment equals the length of tangent segment squared. ( Whole × outside = Tangent squared) - If two chords intersenct inside a circle, then the product of part one and part two on one chord equal the product of part one and part two of the other chord. (Part 1 × Part 2 = Part 1 × Part 2)



__**Chapter 9.6**__

__//Objectives://__
- Create an equation to graph a circle on the origin. - Create an equation to graph a circle not on the origin.

//__Equations:__//
- When the center of the circle is not at the origin. - When the center of the circle is at the origin (0,0)
 * (X - h)² + (Y - k)² = r ²
 * (h,k) is the origin (make sure you use the opposite of the numbers for the origin).
 * X² + Y² = r ²