ࡱ> zx#` Xbjbj\.\. m>D>DE* &&&&RRRRooo8oZqR*t8Jy`y`yzmts<$h%8R7t6m8&&`yzGM666&*7`yRz666P=:R6zt  o6Tc06F6R6Ah6T88̴jRRRdlRRRlRRR&&&&&& 9-1 Basic Terms associated with Circles and Spheres Circle __________________________________________________________________ Given Point = __________________ Given distance = _____________________ Radius__________________________________________________________________ Chord____________________________________________________________________ Secant___________________________________________________________________ Diameter__________________________________________________________________ Tangent___________________________________________________________________ Point of Tangency___________________________________________________________ Sphere____________________________________________________________________ Label Accordingly:     Congruent circles or spheres__________________________________________________ Concentric Circles___________________________________________________________ Concentric Spheres__________________________________________________________ Inscribed in a circle/circumscribed about the polygon________________________________ _______________________________________  HYPERLINK "http://www.pinkmonkey.com/studyguides/subjects/geometry/chap7/g0707101.asp" http://www.pinkmonkey.com/studyguides/subjects/geometry/chap7/g0707101.asp  HYPERLINK "../../../3%20SKETCHPAD%20FOR%20GEOMETRY/CH%209/CIRCLE%20TERMS.GSP" SKETCHPAD 9-2 Tangents  HYPERLINK "../POWERPOINT%20PRESENTATIONS/Chapter%209-2.ppt" POWERPOINT Theorem 9-1 If a line is tangent to a circle , then the line is __________________________ _________________________________.    Corollary: Tangents to a circle from a point are __________________________      Theorem 9-2 If a line in the plane of a circle is perpendicular to a radius at its outer endpoint, then the line is ________________________. Inscribed in the polygon/circumscribed about the circle:     Common Tangent ___________________________________________________ Common Internal Tangent   Common External Tangent   Tangent circles ________________________________________________________ Draw the tangent line for each drawing   Name a line that satisfies the given description.       1. Tangent to ( P but not to ( O. _______ 2. Common external tangent to ( O and ( P. _______ 3. Common internal tangent to ( O and ( P. _______ 4. Circles A, B, C are tangent . AB = 7, AC = 5 CB = 9 Find the radii of the circles.  5. Find the radius of the circle inscribed in a 3-4-5 triangle.   HYPERLINK "../../2%20Geo%209%202004-5/POWERPOINT%20PRESENTATIONS/Chapter%209-2.ppt" PP CONCLUSION 6) Circles O and P have radii 18 and 8 respectively.  EMBED Equation.DSMT4  is tangent to both circles.  Find AB.Hint: connect centers. Find a rt.  9-3 Arcs and Central Angles Central Angle ________________________________________________________ Arc ________________________________________________________________ Measure of a minor arc = ______________ Measure of a major arc = __________ - ______________ Adjacent arcs ____________________ Measure of a semicircle = ___________________ Postulate 16 Arc Addition Postulate: The measure of the arc formed by two adjacent arcs is _________________________________________. That is, arcs are additive. Just like with angles, to differentiate an arc from its measure, an m must be included in front of the arc. Congruent arcs _______________________________ Theorem 9-3 In the same circle or _________________, two minor arcs are _____________ if _________________________________.        1. Name 2. Give the measure of each angle or arc: a) two minor arcs a) AC b) two major arcs b) m(WOT c) a semicircle c) XYT d) an acute central angle e) two congruent arcs 3. Find the measure of (1 (the central angle) a) b) c) d) 4. Find the measure of each arc:  a) AB b) BC c) CD d) DE e) EA 5) a) If  EMBED Equation.DSMT4 , AO = 10, find <1, <2 and AB b) If <2 = x find <1,  EMBED Equation.DSMT4  9-4 Arcs and Chords The arc of the chord is _______________________________________ Theorem 9-5 A diameter that is perpendicular to a chord _______________ the chord and _________________________.  That is, in ( O with CD ( AB, AZ = BZ and AD ( BD How?   Find the following: 1. x = ______ y = ______ 2. x = ______ y = ______ mAB = ______    3. MN = ______ KO = ______ 4. = ______ m(AOC = ______    5. x = ______ y = ______ 6. mCD = ______   7.  EMBED Equation.DSMT4  =  EMBED Equation.DSMT4 , FIND  EMBED Equation.DSMT4  8. If OC = 6, find x and y 9-5 Inscribed Angles By definition, an inscribed angle is an angle whose VERTEX IS ON THE CIRCLE and is contained in the circle. Inscribed angles can intercept a minor arc or a major arc.   Theorem 9-7 The measure of an inscribed angle is equal to ________________________________ Find angle A and angle B. What generalization can you make?  Corollary 1: If two inscribed angles __________________ _____________________________  Corollary 2: An inscribed angle that intercepts a diameter _________________________________ Corollary 3: If a quadrilateral is inscribed in a circle, then its opposite angles are ________________  Theorem 9-8 the measure of an angle formed by a chord and a tangent is equal to ____________ of the intercepted ___________.  Solve for the variable(s) listed:       HYPERLINK "../../2%20Geo%209%202004-5/POWERPOINT%20PRESENTATIONS/Ch%209/9-5%20Inscribed%20Angles.ppt" POWERPOINT 9-6 Other Angles  HYPERLINK "../../../3%20SKETCHPAD%20FOR%20GEOMETRY/CH%209/Circles%20for%20packet.gsp" Sketchpad Theorem 9-9 The measure of an angle formed by two chords that intersect inside a circle is equal to  EMBED Equation.DSMT4  the sum of the intercepted arcs.  That is: ____________________ Theorem 9-10 The measure of an angle formed by secants, two tangents or a secant and a tangent is equal to ______________________________________ THE VERTEX IS OUTSIDE THE CIRCLE Case 1 Case 2 Case 3 2 secants 2 tangents secant/tangent     _________________ _________________ __________________ Given UT is tangent to the circle, m(VUT = 30. Find the following:  1. m EMBED Equation.DSMT4  = ________ 2. m(TVS = ________ 3. m(RVS = ________ 4. m EMBED Equation.DSMT4  = ________  Given the drawing: AB is tangent to ( O; AF is a diameter; m EMBED Equation.DSMT4  = 100, m EMBED Equation.DSMT4  = 30, m EMBED Equation.DSMT4  = 25. Find the measures of angles 1-8.  1= 2= 3= 4= 5= 6= 7= 8= ANGLE MEASUREMENT BASED ON VERTEX 1) VERTEX AT CENTER angle = ______________ 2) VERTEX ON CIRCLE angle = ______________   3) VERTEX INSIDE CIRCLE angle = ______________  4) VERTEX OUTSIDE THE CIRCLE angle = ______________     SECANT/SECANT TANGENT/SECANT TANGENT/TANGENT 9-7 Circles and Lengths of Segments Theorem 9-11 When two ________ intersect inside a circle, the __________ of the _______ of _______ ____________ equals the ___________ of the ______________ of the ___________ ______________. That is, in the circle below, given that the two chords intersect, the equation is    ____________ or __________________________ Theorem 9-12 When two ________ segments are drawn to a circle from an _________ _____________, the product of one secant segment and its __________ ______________ is equal to the product of the other secant segment and its _______________________ That is, in the circle below,  _____________ or _______________________________ Theorem 9-13 When a _______ segment and a _________ segment are drawn to a circle From an ___________ ________ the product of the secant segment and Its _______ _________ is equal to the __________ of the ____________. That is, in the circle below:  _______________ or ____________________________ EXAMPLES:       SKETCHPAD  HYPERLINK "../../2%20Geo%209%202004-5/POWERPOINT%20PRESENTATIONS/Ch%209/Circles%20and%20Lengths%20of%20Segments.ppt" POWERPOINT  Find the measure of each numbered angle given arc measures as indicated. (42 is a central angle m(1__________ m(2__________ m(3__________ m(4___________ m(5___________ m(6__________ m(7__________ m(8__________ m(9___________ m(10__________ m(11_________ m(12_________ m(13_________ m(14__________ m(15__________ m(16_________ m(17_________ m(18_________ m(19__________ m(20__________ m(21_________ m(22_________ m(23_________ m(24__________ m(25__________ m(26_________ m(27_________ m(28_________ m(29__________ m(30__________ m(31_________ m(32_________ m(33_________ m(34__________ m(35__________ m(36_________ m(37_________ m(38_________ m(39__________ m(40__________ m(41_________ m(42_________ m(43_________ m(44__________ m(45__________ CH 9 CIRCLE REVIEW  (1) Find the measure of each of the numbered (2) The three circles with centers A , B , and C angles, given the figure below with arc are tangent to each other as shown below. measures as marked. Point O is the center Find the radius of each circle if AB = 12 ,  of the circle. AC = 10 and BC = 8.  m(1 =____ m(2 =____ m(3 =____ m(4 =____ m(5 =____ m(6 =____ m(7 =____ m(8 =____ Circle A_____ , Circle B_____ , Circle C_____ m(9 =____ m(10 =____  (3)  EMBED Equation.DSMT4 = 120( , AO = 6. Find: AB_____ (4) m(A = 80( Find:  EMBED Equation.DSMT4 ______   (5)  EMBED Equation.DSMT4  is tangent to the circle with center O. (6)  EMBED Equation.DSMT4  is a diameter,  EMBED Equation.DSMT4  , AC = 3 , AB = 2 , OC = 3. Find: BC______ BC = 6. Find: CD______  (7) AE is tangent at B,  EMBED Equation.DSMT4  is a diameter, (8)  EMBED Equation.DSMT4  is a diameter,  EMBED Equation.DSMT4  is tangent at B, m(A = 40(. Find:  EMBED Equation.DSMT4 ____, m( EBD____  EMBED Equation.DSMT4  = 120( , AD =  EMBED Equation.3 .  Find: BC_____, CD_____, OA_____   (9)  EMBED Equation.DSMT4  is tangent at A, AF = FD, sides as marked. (10) Given the figure with sides as marked, Find: EF______ , AF_______ Find: BC_______ , EF_______    (11) Circles with centers O and P as shown, (12) Given the figure below with sides as OP = 15 , OC = 8 , PD = 4 marked, find the radius of the inscribed Find: AB______ , CD_______ circle________   Answers (1) m(1 =  EMBED Equation.DSMT4  , m(2 =  EMBED Equation.DSMT4  , m(3 =  EMBED Equation.DSMT4  , m(4 =  EMBED Equation.DSMT4  m(5 =  EMBED Equation.DSMT4  , m(6 =  EMBED Equation.DSMT4  , m(7 =  EMBED Equation.DSMT4  , m(8 =  EMBED Equation.DSMT4  m(9 =  EMBED Equation.DSMT4  , m(10 =  EMBED Equation.DSMT4  (2) Circle A = 7 , Circle B = 5 , Circle C = 3 (3)  EMBED Equation.DSMT4  (4)  EMBED Equation.DSMT4  (5) BC = 4 (6) CD =  EMBED Equation.DSMT4  (7)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (8) BC =  EMBED Equation.DSMT4  , CD =  EMBED Equation.DSMT4  , OA = 6 (9) EF = 9 , AF = 6 (10) BC = 4 , EF = 8 (11) AB = 9 , CD =  EMBED Equation.DSMT4  (12) 4 CH 9 CIRCLES REVIEW II (1) The circle with center O is inscribed in (ABC. (2) CA is tangent to the circle at A, sides as   EMBED Equation.DSMT4 . Find: AC______ , BC_______ marked. Find: AC_______   (3)  EMBED Equation.DSMT4  is an external tangent segment. Points (4) Concentric circles with center O,  EMBED Equation.DSMT4  is O and P are the centers of the circles. tangent to the inner circle, sides as marked. Find: AB_________ Find: OB_______ ,  EMBED Equation.DSMT4 ________   (5) Given the figure below, point O is the center (6) Given the figure below, m(A = 30(, the circle,  EMBED Equation.DSMT4 , BD = 26 , AC = 24. m(CFD = 65(, BC = DE. Find: OE_____ , DE_____ , OC______ Find:  EMBED Equation.DSMT4 ____,  EMBED Equation.DSMT4 ____,  EMBED Equation.DSMT4 ____   (7) The circle below with center O, AC = 12 , (8) Given the figure below, DH = HF, with  EMBED Equation.DSMT4 . sides as marked. Find: OE______ , OC_______DE_______ Find: GC_______ , DH________    (9) The circle with center O is inscribed (10) Points O and P are the centers of the in (ABC as shown below. AB = AC, circles below. CP = 6 sides as marked. Find: OE_________ Find: AB_______ ,  EMBED Equation.DSMT4 ________    (11) A chord whose length is 30 is in a circle whose radius is 17. How far is the chord from the center of the circle? Review Answers II (1) AC = 6 , BC = 8 (2) AC =  EMBED Equation.DSMT4  (3) AB =  EMBED Equation.DSMT4  (4) OB = 4 ,  EMBED Equation.DSMT4  (5) OE = 5 , DE = 8 , OC = 13 (6)  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (7) OE =  EMBED Equation.DSMT4  , OC =  EMBED Equation.DSMT4  , DE =  EMBED Equation.DSMT4  (8) GC =  EMBED Equation.DSMT4  , DH =  EMBED Equation.DSMT4  (9) OE =  EMBED Equation.DSMT4  (10) AB =  EMBED Equation.DSMT4  ,  EMBED Equation.DSMT4  (11) 8 CH 9 CIRCLES ADDITIONAL REVIEW 1) Find the radius of a circle in which a 48 cm chord is 8 cm closer to the center than a 40 cm chord.  AB = 48, CD = 40  2) In a circle O, PQ = 4 RQ = 10 PO = 15. Find PS.   3) An isosceles triangle, with legs = 13, is inscribed in a circle. If the altitude  to the base of the triangle is = 5, find the radius of the circle. (There are 2 situations) Answers: 1) 25 2) 2 3) 16.9 SUPPLEMENTARY PROBLEMS CH 9 1) Fill out page one of the Circles Packet.  2) A regular polygon is inscribed in a circle so that all vertices of the quadrilateral intersect the circle. What happens to the regular polygon as the number of sides increases. 3) A circle with a center at (2,1) is tangent to the line y = 3x + 5 at A(-1,2). Make a sketch in the coordinate plane and draw a radius from the center of the circle to the radius at point A? Why? 4) In the picture below, AB is a common external tangent. How many common external tangents can be drawn connecting the 2 circles in each of the following pictures? What shape can be formed if a radius drawn to a tangent is perpendicular to the tangent?     5) If the central angle of a slice of pizza is 36 EMBED Equation.DSMT4  degrees, how many pieces are in the pizza? 6) Circle O has a diameter DG and central angles COG = 86, DOE = 25, and FOG = 15. Find the minor arcs CG, CF, EF, and major arc DGF. 7) Draw a circle and label one of its diameters AB. Choose any other point on the circle and call it C. What can you say about the size of angle ACB? Does it depend on which C you chose? Justify your response, please.    8) If two chords in the same circle have the same length, then their minor arcs have the same length, too. True or false? Explain. What about the converse of the statement? Is it true? Why? 9) Draw a circle. Draw two chords of unequal length. Which chord is closer to the center of the circle? What can be said of the intercepted arcs? 10) If P and Q are points on a circle, then the center of the circle must be on the perpendicular bisector of chord PQ. Explain. Which point on the chord is closest to the center?  11) The Star Trek Theorem: a.) Given a circle centered at O, let A,B,and C be points on the circle such that arc AC is not equal to arc BC and CL is a diameter. 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