Tyler's+Chapter+9

9.1
 * __Objectives:__**
 * Define and use the degree measure of arcs.
 * Define a circle and its associated parts, and use them in constructions.
 * Define and use the length measure of arcs.
 * Prove a theorem about chords and their intercepted arcs.
 * __Definition- Circle:__** 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 circle. A **radius** is a segment from the center of the circle to a point on the circle. A **chord** is a segment whose endpoints line on a circle. A **diameter** is a chord that contains the center of a circle.


 * __Definition- Degree Measure of Arcs:__** The degree measure of 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°.


 * __Arc Lenght-__** If //r// is the radius of a circle an //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°(2×3.14r)

Circle:** Set of all poings in a plane that are equidistant from a given point in the plane, center of the circle. __**Intercepted Arc:**__ An arc whose endpoints lie on the sides angle and whose other points lie in the interior of the angle.
 * Definitions***
 * __Radius:__** Segment from the center of the circle to a point on the circle.
 * __Chord:__** A segment whise endpoints line on a circle.
 * __Diameter:__** A chord that contains the center of a circle.
 * __Arc:__** An unbroken part of a circle.
 * __Endpoints:__** Any two distinct points on a circle that divide the circle into two arcs.
 * __Semi-Circle:__** an arc whose endpoints are endpoints of a diameter.
 * __Minor Arc:__** An arc that is shorter than a sami-circle of that circle.
 * __Major Arc:__** An arc that is longer than a semi-circle of that circle.
 * __Central Angle:__** An angle in the plane of a circle whose vertex is the center of the circle.

9.2
 * __Objectives__**
 * **Define:** incribed angle and intercepted arc.
 * **Develop:** and use the inscribed angle theorem and its corollaries.


 * Secant**- Is represented by the line on the bottom that goes right through the circle.
 * Tangent-** Is represented by the line on the top which goes right along the circle.
 * ^^^^Inscribed Angle** - An angle whose vertex lies on a circle and whose sides of the circle.**^^^^_ __Represented by the image above.__**


 * __Blue Boxes__**
 * Inscribed Angle Theorem:** The measure of an inscribed is equal to half the measure of the intercepted arc.
 * Right Angle Corollary:** If an inscribed angle intercepts a simicircle, then the angle is a right angle.
 * Arc-Intercepted Corollary:** If two inscribed angles intercept the same arc, then they have the same measure.
 * Theorem:** The perpendicular bisector of a chord passes through the center of the circle.
 * Radius and Chord Theorem:** A radius that is perpendicular to a chord of a circle.
 * Secant and Tangents:** A secant to a circle is a line that intersects the circle at two points. A tangent is a line in the plane of the circle that intersects the circle at exactly at one point, which is known as the point of tangency.

9.3 Define: Inscribed angle and intercepted arc. Develope and use the inscribed angle theorem and its corollaries.
 * __Objectives:__**

Angle:**An inscribed angle is an angle whose vertex lies on a circle and whose sides are chords of the circle.
 * Definition:
 * Right- Angle Corollary:** If and 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.

9.4
 * Define:** Angles formed by secants and tangents of cirlces.
 * Develope:** and use theorems about measures of arcs intercepted by these angles.

Theorem:** 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 ½ the measure of its intercepted arc.
 * Blue Box.
 * Theorem:** The measure of an angle formed by two secants or chords that intersect in the interior of a circle is 1/2 the sum of the measures of the arcs intercepted by the angle and its vertical angle.
 * Theorem:** The measure of an angle formed by two secants that intersect in the exterior of a circle is 1/2 the different of the measure of the intercepted arcs.
 * Theorem:** 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 1/2 the measure of its intercepted arc.


 * Example:**

Find angle XYZ in each figure. __Circle A:__ Angle XYZ is formed by a secant and a tangent that intersect on the circle. measure of angle XYZ is ½ measure of arc XY (200°) = 100°. __Circle B:__ Angle XYZ is formed by two secants that intersect inside the circle. measure of angle XYZ is ½ (measure of arc XY + measure of arc PQ)1/2 (100° + 20°) = 60°. __Circle C:__ Angle XYZ is formed by two secants that intersect outside the circle. measure of angle XYZ is ½ (measure of arc XZ - measure of arc PQ)1/2 (100° - 50°) =25°.

9.5 Tangent Segment:** a segment that is contained by a line tangent to a circle and has one of its endpoints on the circle.
 * Blue Box.
 * Secant Segment:** a segment that contains a chord of a circle and has one endpoint exterior to the circle and the other endpoints on the circle.
 * External Secant Segment:** the portion of a secant segment that lies outside the circle.
 * Chord:** a segment whose endpoints lie on a circle.
 * Theorem:** If two segments are tangent to a circle from the same external points, then the segments are equal.
 * Theorem:** If two secants intersect outside a circle, the product of the lengths of one secant segment and its external segment equals the prroduct of the lengths of the one secant, (whole * outside = whole * outside)
 * Theorem:** If a secant and a tangent intersect outside a circle, then the product of the lenghts of the segment and its external segment equals tangent squared. ( whole * outside = tangent squared)
 * Theorem:** If two chords intersect inside a circle the nthe product of the lenghts of the segment of one chord equals the product of the length of the chord.


 * Example:**

__Circle A:__ XY = XZ because they are tangents to a circle from the same external point. __Circle B:__ Whole x outside = Whole x outside so AC•BC = EC•DC 8•3 = 6•DC DC = 4 __Circle C:__ Let //d// be the distance to the end. //d// • 50 = 100² //d •// 50 = 10,000