maan56

=Chapter 9=

9.1
__Objections:__ 1) Define a circle and its associated parts, and use them in consructions. 2) Define and use the degree measure of arcs. 3) Define and use the length measure of acrcs. 4) Prove a theorem about chords and their intecepted arcs.

__Definitions:__ 1) __Circle__ - Set of points in a plainthat are equidistance from a given centeral point. 2) __radius__ - A segment that connects the center of a circle with a point on the circle (half the diameter of the circle). 3) __diameter__ - A chord that passes through the center of the circle (two times the length of the radius). 4) __Centeral angle__ - An angle formed by two rays starting from the center of a circle. 5) __Intercepted arc__ - An arc with endpoints that lie inside the inscribed angle. 6) __Minor arc__ - An arc of a circle that is shorter than a cemicircle. 7) __Major arc__ - An arc of a circle that is larger than a cemicircle

__Theorems:__ 1) __Chords and arcs theorem__ - In a circle, or in congruent cicles, the arcs of congruent chords are Congruent. 2) __The converse of the chords and arcs theorem__ - In a circle or in congruent circles, the chords of congruent arcs are congruent.

__Equations:__ 1) __Arc lenght__ - r is the radius of a circle and M is the degree measur of an arc of the circle, then the length, L, of athe arc is given by the following: L = M/630(2πr)

__Example:__ a circle with a diameter of 20 and a centeral angle of 30° Solution - L = 30/360(2 10) L aproximatly 5.235987756

Quiz for 9.1



9.2
__Objections:__ 1) Define tangents and secants of circle. 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.

__Definitions:__ 1) Sectants - A line that intersects a circle at two points. 2) Tangents - In a right triangle, the ratio of the lenght of the side opposite an acute angle to the length of the side adjacent to it. 3) Point of tangency - The point of a circle or sphere with a tangent line or plane.

__Theorems:__ 1) 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. 2) radius and chord theorem - a radius that is perpendicular to a chord of a circle is bisect of the chord. 3) Converse of the tangent theorem - If a line is perpendicular to a radius of a circle at its endpoints 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.

//Quiz for 9.2//

9.3
__Objections:__ 1) Define inscribed angle and intercepted arc 2) Develop and use the inscribed angle theorem and its corollaries.

__Definitions:__ 1) Inscribed angle - An angle whose vertex lies on a circle and whose sides are cords of the circle.

__Theorems:__ 1) Inscribed angle theorem - The measure of an angle inscirbed in a circle is equil 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.

//Quiz for 9.3//



9.4
__Objections:__ 1) Define angles fromed by secants and tangents of circles. 2) Develope and use theorems about measures of arcs intercepted by thse angles.

__Other things to know:__ 1) Vertex is on the circle a) Secant and tangent b) Two secants 2) Vertex is inside the circle a) Two secants 3) Vertex is outside the circle a) Two tangents b) Two secants c) Secant and tangent

__Theorems:__ 1) Theorem 9.4.1 - If a tangent and a secant (or a chord) intersect on a circle at the point of tangency, then the measure of the angle is formd is one half the measure of its intecepted arc. 2) Theorem 9.4.2 - The measur of an angle fromed by two secants or chords that intersect in the interior of a circle is one half the sum of the mesures of the arcs intercepted by the angle and its wertical angle. 3) Theorem 9.4.3 - The measur eof 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. 4) Theorem 9.4.5 - 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. 5) Theorem 9.4.6 - 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.

//Quiz for 9.4//

9.5
__Objections:__ 1) Define special cases of segments related to circlesm including secant secant, secant tangent, and chord chord segments. 2) Develop and se theorems about measures of the segments.

__Definitions:__ 1) Tangent segment - A segment that is contained by a line tangent to a circle and has one of its endpoints on the circle. 2) 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. 3) External secant segment - The potion of a secant segment theat lies outside the circle. 4) Chord - A segment whose endpoints lie on a circle.

__Theorems:__ 1) If two secants are tangent to a circle fom the same external point, then the segments are of equil length. 2) If two secants intersect outside a circle, then the product of the lengths of one secant segment and its extenal segment equials the product of the lenghts of the tangent segment squared (whole X outside = whole X outside) 3) If a secant and a tenagent intersects outside a circle, then the priduct of the lengthsof a secant segment and its external segment equils the length of the tangent segment squeared (whole X outside = Tangent squared) 4) If two chords intersect inside a circle then the priduct of the lenghts of the segment of none chord equals the product of the lenght of the sengments of the other chord.

//Quiz for 9.5//



9.6
objections: 1) Develope and use the equation of a circle. 2) Adjust the equation for a circle to move the center in a coodinate plane.

Equations: 1) X^2 + Y^2 = r^2 2) (x - h)^2 + (y - k)^2 = r^2

Example: 1) there is a circle on the origin (0,0) and a radius of 5 So x^2 + Y^2 = 5^2 2) There is a circle with a center of (5,4) and a radius of 3 So (x - 5)^2 + (y - 4)^2 = 3^2

//Quiz for 9.6//

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