joam321

Chapter 9


-Define a circle and its associated parts and use them in consruction. -Define and use the degree measure of arcs. -Define and use the length measure of arcs. -Prove a theorem about chords and their intercepted arcs.
 * 9.1**
 * __Objectives__**

__Circle__ - Set of points in a plain that are equal distance from a given central point. __radius__ - A segment that connects the center of a circle with a point on the circle [half the diameter of the circle]. __diameter__ - A chord that passes through the center of the circle [two times the length of the radius]. __Centeral angle__ - An angle formed by two rays starting from the center of a circle. __Intercepted arc__ - An arc with endpoints that lie inside the inscribed angle. __Minor arc__ - An arc of a circle that is shorter than a semicircle. __Major arc__ - An arc of a circle that is larger than a semicircle __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 http://www.flickr.com/photos/scragz/134087192/
 * __Definitions:__**

__**Example:**__ a circle with a diameter of 50 and a central angle of 40° L = 40/360(2pi 25) L= 17.45

9.2
__**Objectives**__ -Define tangents and secants of a circle. -Understand the relationship between tangents and certain radii of circles. -Understand the geometry of a radius perpendicular to a chord of a circle.

__**Definitions:**__ __Sectants__ - A line that intersects a circle at two points. __Tangents__ - In a right triangle, the ratio of the length of the side opposite an acute angle to the length of the side adjacent to it. __Point of tangency__ - The point of a circle or sphere with a tangent line or plane.

__**Theorems:**__ __Tangent Theorem__ - If a line is the tangent to 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 is bisect of the chord. __Converse of the tangent theorem__ - A line is perpendicular to a radius of a circle at its endpoints on the circle, then line is tangent to the circle. __Theorem__ - The perpendicular bisector of a chord passes through the center of the circle.

http://www.flickr.com/photos/88261881@N00/365455942/

Q: What do you use to find a measure of a chord if you know the chord perpendicular to it? A: The Pythagorean Theorem.
 * __Example__**

9.3
photo by Bas van Dijk Objectives__** -Define inscribed angle and intercepted arc -Develop and use the inscribed angle theorem and its corollaries.
 * __[[image:ferris_wheel.jpg link="http://www.flickr.com/photos/basvandijk/427113517/"]]

__**Definitions:** Inscribed angle__ - An angle whose vertex lies on a circle and whose sides are cords of the circle. [|link to inscribed angles]

__Inscribed angle theorem__ - The measure of an angle inscirbed 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 2 inscribed angles intercept the same arc, then they have the same measure. [|an arc used in everyday life]
 * __Theorems:__**

Find the measure of <JKL. <JKL is inscribed in S and intercepts arc JL. m<JKL (1/2)m(arc)JL (1/2)(45) = 221/2
 * __Example:__**

9.4
__**Objections:**__ -Define angles fromed by secants and tangents of circles. -Develope and use theorems about measures of arcs intercepted by thse angles.

__Vertex is on the circle__ -Secant and tangent -2 secants __Vertex is inside the circle__ -2 secants __Vertex is outside the circle__ -2 tangents -2 secants -Secant and tangent [|helpful site lots of information]
 * __Definitions:__**

__**Theorems:**__ __Theorem 9.4.1__ - If a tangent and a secant intersect on a circle at the point of tangency, then the measure of the angle is formed is one half the measure of its intecepted arc. __Theorem 2__ - The measure of an angle fromed 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. __Theorem 3__ - The measure of an angle formed by two secants that intesect in the exterior of a circle is one half the difference of the measure of the intercepted arcs. __Theorem 5__ - The measure of a secant tangent angle with its vertex outside the circle is one half the difference of the measure of the intercepted arcs. __Theorem 6__ - The measure of a tangent tangent angle with its vertex outside the cirlcle is one half the difference of the measures of the intercepted arcs, or the measre of the mahor arc minus 180 degrees.


 * Picture Made by me**

__**Example**__ If x1=50 and x2=70 50+70=120/2 So your answer is 60

__**Objections:**__ -Define special cases of segments related to circlesm including secant secant, secant tangent, and chord segments. -Develop and se theorems about measures of the segments.
 * 9.5**

__**Definitions:**__ __Tangent segment__ - A segment that is contained by a line tangent to a circle and has one of its endpoints on the circle. __Secant segment__ - A segment that contains a chord of a circle and has one end point exterior to the circle and the other endpoint 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.

__**Theorems:**__ -If two secants are tangent to a circle fom the same external point, then the segments are of equal length. -If two secants intersect outside a circle, then the product of the lengths of one secant segment and its extenal segment equals the product of the lengths of the tangent segment squared. -If a secant and a tangent intersects outside a circle, then the product of the lengths of a secant segment and its external segment equals the length of the tangent segment squared. -If two chords intersect inside a circle then the product of the lengths of the segment of none chord equals the product of the length of the segments of the other chord.

__**Example**__ If segment AC is 20 and EC is 5, BD is 5 what is DE? 20*5=25*x 100=25x 100/25=25x/25 X=4

9.6
__**Objections:**__ -Develope and use the equation of a circle. -Adjust the equation for a circle to move the center in a coordinate plane.

A circle on the origin (0,0) and a radius of 8 x^2 + Y^2 = 8^2 A circle with a center of (2,7) and a radius of 4 (x - 2)^2 + (y - 7)^2 = 4^2
 * __Example:__**