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Circles. Chapter 10. 10.1 Tangents to Circles. Circle : the set of all points in a plane that are equidistant from a given point. Center : the given point. Radius : a segment whose endpoints are the center of the circle and a point on the circle. Vocabulary.

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CirclesChapter 1010.1 Tangents to CirclesCircle: the set of all points in a plane that are equidistant from a given point. Center: the given point. Radius: a segment whose endpoints are the center of the circle and a point on the circle. VocabularyChord: a segment whose endpoints are points on the circle. Diameter: a chord that passes through the center of the circle. Secant: a line that intersects a circle in two points. Tangent: a line in the plane of a circle that intersects the circle in exactly one point. More VocabularyCongruent Circles: two circles that have the same radius. Concentric Circles: two circles that share the same center. Tangent Circles: two circles that intersect in one point. Tangent TheoremsIf a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency. PQTangent TheoremsIn a plane, 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. PQTangent TheoremsIf two segments from the same exterior point are tangent to a circle then they are congruent. QPExamplesFind an example for each term: Center Chord Diameter Radius Point of Tangency Common external tangent Common internal tangent SecantExamplesThe diameter is given. Find the radius. d=15cm d=6.7in d=3ft d=8cm ExamplesThe radius is given. Find the diameter. r = 26in r = 62ft r = 8.7in r = 4.4cm ExamplesTell whether AB is tangent to C. A145B15CExamplesTell whether AB is tangent to C. A12C168BExamplesAB and AD are tangent to C. Find x. D2x + 7A5x - 8B10.2 Arcs and ChordsAn angle whose vertex is the center of a circle is a central angle. If the measure of a central angle is less than 180 , then A, B and the points in the interior of APB form a minor arc.Likewise, if it is greater than 180, if forms a major arc.Arcs and ChordsIf the endpoints of an arc are the endpoints of a diameter, then the arc is a semicircle. The measure of an arc is the same as the measure of its central angle. ExamplesFind the measure of each arc. MN MPN PMN Arc Addition PostulateThe measure of an arc formed by two adjacent arcs is the sum of the measures of the two arcs. mABC = mAB + mBC ExamplesFind the measure of each arc. GE GEF GF TheoremsIn the same circle, or in congruent circles, congruent chords have congruent arcs and congruent arcs have congruent chords. In the same circle, or in congruent circles, two chords are congruent iff they are equidistant from the center. ExamplesDetermine whether the arc is minor, major or a semicircle. PQ SU QT TUP PUQ ExamplesKN and JL are diameters. Find the indicated measures. mKL mMN mMKN mJML ExamplesFind the value of x. Then find the measure of the red arc. HomeworkPg. 600 # 26-28, 37, 39, 47, 48 Pg. 607 # 12-30 even, 32-34, 37-38 10.3 Inscribed AnglesAn inscribed angle is an angle whose vertex is on a circle and whose sides contain chords of the circle. The arc that lies in the interior of an inscribed angle and has endpoints on the angle is called the intercepted arc. Measure of an Inscribed AngleIf an angle is inscribed in a circle, then its measure is half the measure of its intercepted arc. mADB = ½ mAB Find the measure of the blue arc or anglemQTS = m NMP = TheoremIf two inscribed angles if a circle intercept the same arc, then the angles are congruent. Polygons and circlesIf all of the vertices of a polygon lie on a circle, the polygon is inscribed in the circle, and the circle is circumscribed about the polygon. TheoremsIf a right triangle is inscribed in a circle, then the hypontenuse is a diameter of the circle. B is a right angle iff AC is a diameter of the circle. TheoremsA quadrilateral can be inscribed in a circle iff its opposite angles are supplementary. D, E, F, and G lie on some circle, C, if and only if mD + mF = 180° and mE + mG = 180° ExamplesFind the value of each variable. More ExamplesPg 616 # 2-8 10.4 Other Angle Relationships in CirclesIf a tangent and a chord intersect at a point on a circle, then the measure of each angle formed is one half the measure of its intercepted arc. m 1 = ½ mAB m 2 = ½ mBCA ExamplesLine m is tangent to the circle. Find the measure of the red angle or arc. ExamplesBC is tangent to the circle. Find m CBD. TheoremsIf two chords intersect in the interior of a circle, then the measure of each angle is one half the sum of the measures of the arcs intercepted by the angle and its vertical angle. x = ½ (mPS + mRQ) x = ½ (106 + 174 ) x = 140 TheoremsBAm 1 = ½ (mBC – mAC) m 2 = ½ (mPQR – mPR) m 3 = ½ (mXY – mWZ) 1CP2QRXW3ZYMore ExamplesPg. 624 #2-7 10.5 Segment Lengths in CirclesWhen 2 chords intersect inside of a circle, each chord is divided into 2 segments, called segments of a chord. When this happens, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. ExamplesFind x. VocabularyPS is a tangent segment because it is tangent to the circle at an endpoint. PR is a secant segment and PQ is the external segment of PR. TheoremsEA EB = EC ED BAECDTheorems(EA)2 = EC ED EA is a tangent segment, ED is a secant segment. AECDExamplesFind x. ExamplesFind x. x ___ = 10 ___X2 = 4 ____Examples10.6 Equations of CirclesYou can write the equation of a circle in a coordinate plane if you know its radius and the coordinates of its center. Suppose the radius is r and its center is at ( h, k) (x – h)2 + (y – k)2 = r2 (standard equation of a circle) ExamplesIf the circle has a radius of 7.1 and a center at ( -4, 0), write the equation of the circle. (x – h)2 + (y – k)2 = r2 (x – -4)2 + (y – 0)2 = 7.12 (x + 4)2 + y2 = 50.41 ExamplesThe point (1, 2) is on a circle whose center is (5, -1). Write the standard equation of the circle. Find the radius. (Use the distance formula) . . . (x – 5)2 + (y – -1)2 = 52 (x – 5)2 + (y +1)2 = 25 Graphing a CircleThe equation of the circle is: (x + 2)2 + (y – 3)2 = 9 Rewrite the equation to find the center and the radius. (x – (-2))2 + (y – 3)2 = 32 The center is (-2, 3) and the radius is 3. Graphing a CircleThe center is (-2, 3) and the radius is 3. ExamplesDo Practice 10.6C or B together.

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