cucr118

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OBJECTIVES
__**Definition of a diameter:**__ An arc whos endpoints lie on the sides of the angle whose other points lie in the interior of the angle.
 * Define a circle + its associated parts, + use them in constructions.
 * Define + use the degree measure of arcs.
 * Define + use the length of measure of arcs.
 * Prove a theory about chorde + their intercepted arcs.
 * __Definition of a a circle:__** A circle is the set of all points in a plane that are = from a given point in the plane known as the center of the circle.
 * __Definition of a radius:__** A segment from the center of the circle to a point on the circle.
 * __Definition of a diameter:__** A chord that contains the center of a circle.
 * __Definition of a diameter:__** A circle is an angle in the plane of a circle whose vertex is the center of the circle.

The DM of a minor arc is the measure of its central angle. The DM of a major arc is 360D(degrees) - the DM of its minor arc. The DM of a semicircle is 180D. __Solution__ mAB=100D mBC=90D mABC=mAB+mBC=50D+60D=110D If //r// is the radius of a circle and M is the degree measure of an arc of the circle, then L (length) is given by the following: **L=__M__ (2pie//r// )** __CHORDS + ARCS THEOREM__** In a congruent circle the arcs of the chords are congruent.
 * __DEGREE MEASURE OF ARCS__**
 * EX__.)__** Find the measure of AB, BC, + ABC.
 * __ARC LENGTH__**
 * 360

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OBJECTIVES

 * Define tangents + secants of circles.
 * Understand the relationship between tangents + certain radii of circles.
 * Understand the geometry of a radius perpendicular to a chord of a circle.
 * __Definition of a secant:__** A line that intersects the circle at 2 points.
 * __Definition of a tangent:__** A line in the plane of the cirlce that intersects the circle at exactly 1 point. That point is A.K.A. The Point of Tangency.

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. A radius that is perpendicular to a chord of a circle bisects the chord. (AX)² + 6² = 10² (AX)² = 10² - 6² (AX)² = 64 AX = 8 If a line is perpendicular to a radius of a circle at its endpoint on the circle, then the lin is tangent to the circle.
 * __TANGENT THEOREM__**
 * __RADIUS + CHORD THEOREM__**
 * __Ex.)__** Circle P has a radius of 10in and PX is 6in. Segment PR is perpendicular to Segment AB at point X. Find AB?
 * __Solution__** By the Pythagorean Theorem:
 * __CONVERSE OF THE TANGENT THEOREM__**

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OBJECTIVES

 * Define inscribed angle + ntercepted arc.
 * Develop + use the inscribed angle theorem + its corollaries.


 * __Definition of a inscribed angle:__** Angle whose vertex lies on a circle + whose sides are chords of the circle.

The measure of angle inscribed in a circle is equal to half the measure of the intercepted arc.
 * __INSCRIBED ANGLE THEOREM__**

<ABC is inscribed in circle P + intercepts AC. By the inscribed angle theorem: m<ABC ½ mAC ½(80°) = 40°. If an inscribed angle intercepts a semicircle, then the angle is a right angle. If 2 inscribed angles intercept the same arc, then they have the same measure.
 * __Ex.)__** Find the measure of <ABC. This is circle P.
 * __RIGHT ANGLE COROLLARY__**
 * __ARC INTERCEPT COROLLARY__**

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OBJECTIVES

 * Define angles formed by secants + tangents of circles.
 * Develop + use theorems about measures of arcs intercepted by these angles.

__**THEOREM**__ If a tangent + a secant (or a chord) intersects on a circle at the point of tangency, then the measure of the angle formed is half the measure of its intercepted arc. __**THEOREM**__ The measure of an angle formed by 2 secants that intersect in the exterior of a circle is half the different of the measures of the intercepted arcs. __**THEOREM**__ The measure of an angle formed by two secants or chords that intersect in the exterior of a circle is half the sum of the measures of the arcs intercepted by the angle and its vertical angle. __**Ex.)**__ Find m<ABC(use 100 for problem) if the circle is formed by a secant + tangent that intersect on the circle. <ABC ½ (100) 50.

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**OBJECTIVES**
If 2 segments are tangent to a circle from the same external point, then the segments are of = length. If 2 secants intersect outside of a circle, the product of the lengths of one secant segment and its external segment the length of the other external segment and its measures. (Whole x Outside Whole x Outside). If a secant and a tangent intersect outside a circle, then the product of the lengths of the secant and its external segment and its external segment the length of the tangent secant squared. (Whole x Outside Tangent Squared). If 2 chords intersect inside a circle, then the product of the lenghts of the segments of one chord = the products of the lengths of the other chords squared. AB and CB are tangents to a circle from the same external point. So they are = 15,000 miles. = = =9.6=
 * Define special cases of segments related to circles, including secant, secant-tangent, and chord-chord segments.
 * Develop + use theorems about measures of the segments.
 * __THEOREM__**
 * __THEOREM__**
 * __THEOREM__**
 * __THEOREM__**
 * __THEOREM__**
 * __Ex.)__** Satellites are used in navigation. If range of the satellite, AB, is 15000 miles, what is CB?

OBJECTIVES

 * Develop + use the equation of a circle.
 * Adjust the equation for a circle to move the center in a coordinate plane.

x² + 0² = x² = 16 x = ± 4.
 * __Ex.)__** x² + y² = 16