yansta25

__**CHAPTER 9 : CIRCLES** __

[|Circles Are Awesome] [|Circle Theorems]

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 theorem about chords and their intercepted arcs

//vocabulary -//
 * 1) 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
 * 2) radius : segment from the center of the circle to a point on the circle
 * 3) chord : segment whose endpoints line on a circle
 * 4) diameter : a chord that contains the center of a circle
 * 5) arc : an unbroken part of a circle
 * 6) endpoints : any two distant points on a circle that divide the circle into two arcs
 * 7) semicircle : an arc whose endpoints are endpoints of a diameter
 * 8) minor arc : an arc that is shorter than a semicircle of that circle; named by its endpoints major arc : an arc that is longer than a semicircle of that circle; named by its endpoints and another point that lies on the arc
 * 9) 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 angle and whose other pints lie in the interior of the angle
 * 10) degree measure of arcs : the degree measure of a minor arc is the measure of its central angel; the degree measure of a major arc is 360 minus the degree meausre of its minor arc; the degree measure of a semicircle is 180

//theorems -//
 * 1) chords and arcs theorem : in a circle, or in congruent circles, the chords of congruent arcs are congruent


 * __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°(2πr)//**

//examples// - using the figure below, answer the following questions Find the measure of arc CAB and arc CB.
 * arc CAB = arc CA + arc AB
 * arc CAB = 130 + 70
 * arc CAB = 200
 * arc CB = 360 - arc CAB
 * arc CB = 360 - 200
 * arc CB = 160

Find the length of arc AB and round to nearest hundredth.
 * L=M/360°(2πr)
 * L = 130/360(2π3)
 * L = (.361)(18.85)
 * L = 6.80

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

//vocabulary -//
 * 1) secant : a line that intersects the circle at two points
 * 2) tangent : a line in the plane of the circle that intersects the circle at exactly one point
 * 3) point of tangency : the point of intersection of a circle or sphere with a tangent line or plane

//theorems -//
 * 1) 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
 * 2) radius and chord theorem : a radius that is perpendicular to a chord of a circle bisects the chord
 * 3) 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
 * 4) theorem : the perpendicular bisector of a chord passes through the center of the circle

//examples// - using the figure below, answer the following questions (*__Pythagorean Theorem__* can be helpful : **a**//**²+ b² = c²**)// Circle P has a radius of 5in, and PX is 3 in. segment PR is perpendicular to segment AB at point X. Find AX and AB.


 * (AX)² + 3² = 5²
 * (AX)² = 5² - 3²
 * (AX)² = 16
 * AX = 4

Line KL is tangent to circle M at K. if KM is 5 and LM is 12, find KL.
 * 5² + 12² = KL²
 * 169 = KL²
 * 13 = KL

9.3 : Inscribed Angles and Arcs //objectives -//
 * define inscribed angle and intercepted arc
 * develop and use the inscribed angle theorem and its corollaries

//vocabulary -//
 * 1) inscribed angle : an angle whose vertex lies on a circle and whose sides are chords of the circle

//theorems/corollaries -//
 * 1) Inscribed angle theorem : the measure of an angle inscribed in a circle is equal to one-half the measure of the intercepted arc
 * 2) Right angle corollary : if an inscribed angle intercepts a semicircle, then the angle is a right angle
 * 3) Arc-intercept corollary : if two inscribed angles intercept the same arc, then they have the same measure

//examples -// angle ABC is inscribed in circle X and intercepts arc AC. use the inscribed angle theorem to figure what the measure of angle ABC is.
 * measure angle ABC = 1/2(measure arc XY)
 * measure angle ABC = 1/2(50 degrees)
 * measure angle ABC = 25 degrees

9.4 : Angles Formed by Secants and Tangents //objectives -//
 * define angles formed by secants and tangents of circles
 * develop and use theorems about measures of arcs intercepted by these angles

//theorems -//
 * 1) theorem 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 formed is one-half the measure of its intercepted arc
 * 2) theorem 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 angles
 * 3) theorem 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
 * 4) theorem 4 : the measure of a secant-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs
 * 5) theorem 5 : the measure of a tangent-tangent angle with its vertex outside the circle is one-half the difference of the measures of the intercepted arcs, or the measure of the major arc minus 180 degrees

//examples// - find the measure angle of AVC
 * measure angle AVC = 1/2 (measure arc AB + measure arc CD)
 * measure angle AVC = 1/2 (60 degrees + 40 degrees)
 * measure angle AVC = 50 degrees

9.5 : Segments of Tangents, Secants, and Chords //objectives -//
 * define special cases of segments related to circles, including secant-secant, secant-tangent, and chord-chord segments
 * develop and use theorems about measures of the segments

//vocabulary -//
 * 1) tangent segment : a segment that is contained by a line tangent to a circle and has one of its endpoints on circle
 * 2) secant segment : a segment that contains a chord of a circle and ahs one endpoint exterior to the circle and the other endpoint on circle
 * 3) external secant segment : the portion of a secant segment that lies outside the circle

//theorems -//
 * 1) theorem 1 : if two segments are tangent to a circle from the same external point, then the segments are of equal length
 * 2) theorem 2 : if two secants intersect outside a circle then the product of the lengths of one secant segment and its external segment equals the product of the lengths of the other secant segment and its external segment
 * 3) theorem 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 length of the tangent segment squared
 * 4) theorem 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 the other chord
 * __two tangents__* - are **equal**
 * __two secants__* - **W x O = W x O**
 * __one secant one tangent__* - **W x O = t//²//**
 * __two chords__* - **p + 1 x p + 2 = p + 1 x p + 2**

//examples -// using the figure below, find X
 * X x 2 = 8 x 4
 * 2X = 32
 * X = 16

9.6 : Circles in the Coordinate Plane //objectives -//
 * develop and use the equation of a circle
 * adjust the equation for a circle to move the center in a coordinate plane


 * __equation of a circle with center at (0,0)__* : **//x²+y² = r²//**
 * __equation of a circle with center not on origin (h,k)__* : //**(x-h)² +(y-k)² =r²**

examples -// write the equation and find the x- and y-intercepts for each circle
 * x²+y² = 4
 * (2,0) (-2,0) (0,2) (0,-2)
 * (x-5)² +(y-4)² =9²
 * (8,4) (2,4) (5,7) (5,1)