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=Chapter 9=

=9.1=

study of chords and arc
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Definitions radius- segment that starts in the center of the circle and ends on a point on circle chord- a segment which has an endpoint on a circle diameter- a chord that contains the center of the circle central angle- an angle from the plane which the vertex is in the center of the circle intercepted arcs- endpoint lies on the side of the angle and the other points lie on the interior of the angle degree measure of arcs: Minor arc- an arc that is smaller than 180 degrees. Major arc- an arc that is bigger than 180 degrees Semicircle- is half a cicle equal to 180 degrees.

Theorems chords and arc theorems - in a circle, or in congruent circles, the arc of congruent chords are congruent Converse of the chords and arc theorem- in a circle, or in a congruent cicles, the chords of congruent arcs are congruent.

example- if their is a circle with a arc of RT= 90 and another arc of ST= 100 What does RTS equals? RT=90 + ST=100 equals RTS=190

=9.2=

Tangent to circles
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Definitions Secants-a line that intersects a circle at 2 points. Tangent- a line in a plane of the cicle that intersects the circle at exactly 1 point.

Theorems Theorem- the perpendicular bisecter of a chord passes throught the center of the cicle. tangent theorem- if a line is tangent to a circle, then the line is perpendicular to a radius of the cicle drawn to the point of tangency. radius and chord theorem- a radius that is perpendicular to a chord of a circle bisect the chord. 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.

example- if a line is tangent to a circle then it is where on the circle? if it is tangent then it lies on the circle. =9.3=

inscribing angles and arcs
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example- in the figure above find the fraction of the intercepted arc if the purple area equals 35 degrees? A= 35/360

Definitions inscribed angle- a angle whose vertex lies on a circle and whose sides are chords of a circle. 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, they have the same measure.

theorems inscribed angle theorem- the measure of an angle inscribed in a circle is equal to 1 half the intercepted arc. right angle corollary- if an inscribed angle intercepts a semicircle, then the angle is a right angle. arc intercepts corollary- if 2 inscribed angles intercepts the same arc, then they have the same measurement

=9.4=

angles formed by secants and tangents
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example- what is the radius of the circle if OP= 30 and AP= 90? A=94.86

theorem Theorem- if a tangent or a secant intersects on a circle at the point of tangency, then the angle formed is one halfed the measure of its intercepted arc. theorem- the measure of the angle formed by 2 secants or chords that intersects in a interior of a circle is one half the sum of the measure of the arc intersecepted by the angle and its verticle angle. Theorem- the measure of the angle formed by 2 secants that intersects in the exterior of the circle is one half the difference of the measure of the intercepted arcs. theorem- the measure of a secant-tangent angle with its vertex outside the circle is one half the difference of the measure of the intercepted angle. theorem- the measure of the tangent-tangent angle with its vertex outside is one half the difference of the measure of the intercepted arc, or the measure of the major arc minus 180 degrees.

=9.5=

segments of tangents, secants, and chords
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example- from the theorem whole times outside time whole times outside ex times gx=fx times hx 1.31 times .45=1.46 times hx 1.46 times hx= .5895 hx = .40

Theorem theorem-if 2 secants are tangent to a cicle from the same external point, then the segment are of equal length theorem- if 2 secants intersects outside a circle then the product of the lengths of one secant segment and its external segment equals the product of the length of the other secant segment and its external segments. theorem- if a secant and a tengent intersects a 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. theorem- if 2 chords intersect inside a circle then the product of the length of the segment of one chord equals the product of the length of the segment of the other chord.

=9.6=

circles in the cordinate plane
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example- when graphing a equation it is helpfull to find the intercepts. it is eisier to find the intercepts if one of the x or y is 0 X2+O2 =25 X2=25 X=+ or - 5

helpful hints- if x is 4 and y is -3 then when you write the equation it is the opposite of the of the intercepts listed such as (X-4)2+(Y+3)2= Radius

Cool math link http://www.regentsprep.org/Regents/math/geometry/GP15/CircleAngles.htm http://earthmath.kennesaw.edu/main_site/review_topics/trig_functions.htm