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Geometry Chapter 9 Circle Vocabulary Arc Length Angle & Segment Theorems with Circles Proofs Chapter 9: Circles Date Due Section Topics Assignment Written Exercises Definitions Tangents Circumscribed vs. Inscribed Pg. 335(bottom)-337 #1-7, 8(not d), 10, 14-20 even Common Tangent Tangent Circles Arcs (minor and major) Central <’s Arc Addition Postulate Congruent Arcs Length of an Arc Arcs and Chord Relationships Pg. 347 # 1-9, 12, 18, 22 Pg. 354-355 #2-8, 19-21, Pg. 359-361 #1-10, 12-24 even Inscribed Angles Angles formed by Chords, Tangents and Secants Lengths of Segments in a Circle Proofs Study For Test Extra Practice 9.1 9.2 9.3 9.4 9.5 9.6 9.7 More Proofs Review Chapter 9 Worksheet (pg330 classroom ex.all) Pg. 331 #4, 6, 7, 12-15, 17 Pg.337 #1-3 (mixed review-bottom of page) Pg. 341 (bottom)- 342 # 1-6, 10, [note m<CAO = m<2] , #16 Pg. 364 (bottom)- 366 #2-8 even, 14-22 even Worksheet Pg.349 (Self Test 1) #1-6 Pg.367 (Self Test 2) #1-8 Pg.369-370 (Chpt Rev) #1-24 Pg.371 (Chpt Test) #1-18 1 Circle Introductory Vocabulary Geometry Name _______________________ Date ___________ Block _______ Use appropriate notation to name the following in the given diagram. Write a short explanation or definition as needed. circle: center: diameter: radius: chord: arc: semicircle: major arc: minor arc: ________________________________________________________________________ secant: tangent: ________________________________________________________________________ inscribed polygon: circumscribed polygon: 2 p.330 Class Exercises 1. Name three radii of O. 2. Name a diameter. 3. Consider RS and RS . Which is a chord and which is a secant? 4. Why is TK not a chord? 5. Name a tangent to O. 6. What name is given to point L? _______________________________________________________________ 7. Name a line tangent to sphere Q. 8. Name a secant of the sphere and a chord of the sphere. 9. Name 4 radii. (none are drawn on the diagram) ________________________________________________________________ 10. What is the diameter of a circle with radius 8? 5.2? 4 3 ? j? 11. What is the radius of a sphere with diameter 14? 13? 5.6? 6n? __________________________________________________________________ The radius of circle O has a length of 20. Radii OA and OB are drawn in, forming an angle with the given measure. Find the length of AB using your knowledge of isosceles and special triangles. a) m<AOB = 90 b) m<AOB = 60 c) m<AOB = 120 3 ________________________________________________________________________ 9-3: Arcs and Central <'s & 11-6: Arc Length An arc is measured in degrees - Its measure is equal in measure of the central angle which intercepts it. Arcs are ≅ iff. their central <'s are ≅, with angles 0<θ<360. 4 The central angles are equal in measure.... While the arcs are equal in measure, the arcs are different in length! Arc length is related to circumference.... C = πd or C = 2 π r Arc length.... l = l = central measure 360 central measure 360 d 2 r Think about it -- arc length is a fractional part of the circumference of the circle & the circle is 360 degrees!!!! ________________________________________________________________________ Thm: the measure of a central angle = the measure of the intercepted arc 5 Central Angle & Arcs Notes Geometry Name _________________________ Date ________ Block _________ Find the measure of each arc in the diagram. ________________________________________________________________________ Use the diagram to answer the following: ________________________________________________________________________ 6 Arc Length Practice WS Determine the length of an arc with the given central angle measure, m<M, in a circle with radius r. Give your answers in simplest form in terms of . ____________________________________________________________________ Determine the length of an arc with the given central angle measure, m<M, in a circle with radius r. Give your answers rounded to the nearest hundredth. ___________________________________________________________________ Determine the degree measure of an arc with the given length, L, in a circle with radius r. Give your answers rounded to the nearest tenth. Extra Practice: p.341 CE(1-13) 7 Thm: A line tangent to a circle the line is perpendicular to the radius @ the pt of intersection Converse: A line which is perpendicular to the radius @ a point on the circle the line is tangent to the circle * the circle & line must be Coplanar! Thm: parallel lines/chords in a circle intercepted arcs are congruent Thm: Tangent segments from an external point are congruent Thm: If a line in the plane of a circle is perpendicular to the radius (diameter) at its outer endpoint, then the line is tangent to the circle. ___________________________________________________________________ 8 3. If AP = 12 and BP = 6, find AO. ______________________________________________________________________ Tangents with Circles Internal Tangent Line External Tangent Line Tangent Circles _______________________________________________________________________ 9 Thm: In the same circle or congruent circles, congruent arcs congruent chords Thm: If a diameter of a circle is perpendicular to a chord it bisects both the chord and its intercepted arc. Converse: if diameter bisects a chord it is perpendicular to the chord @ its midpoint Thm: In the same circle or congruent circles, If 2 chords are equidistant from the center the chords are congruent. Converse: if chords are congruent the chords are equidistant from the center ________________________________________________________________________ 10 Classwork: 1) p.349 ST1 (1-6) 2) p.346 Class Exercises (1-6) _______________________________________________________________________ Thm: the measure of an inscribed angle = half the measure of the intercepted arc * 2 inscribed angles which intercept the same arc angles are congruent * An angle inscribed in a semicircle right angle Proof of theorem on next page… 11 ________________________________________________________________________ Corollary 1: 2 inscribed angles which intercept the same arc are congruent. 12 Corollary 2: An angle inscribed in a semicircle is a right angle. Corollary 3: quadrilateral inscribed in a circle opposite angles are supplementary Thm: an angle formed by a tangent & a chord its measure = half the measure of the intercepted arc ________________________________________________________________________ Examples (p.353 #4 – 9) Find the value for x and y in each question. 13 Inscribed Angles Notes Geometry Name ___________________________ Date __________ Block ___________ ______________________________________________________________________ 14 Angles formed by a tangent and a chord – Notes ________________________________________________________________________ 15 Thm: If 2 chords intersect inside a circle the products of the segments on each chord are equal Proof of theorem: 16 Thm: an angle formed by 2 secants/tangents its measure = half the difference of the measures of the intercepted arcs ________________________________________________________________________ p.359 CE(1-10) 6 17 Other Angle Relationships Geometry Name _______________________ Date _____________ Block _____ ___________________________________________________________________ 18 19 Circle Formulas Geometry Name______________________ Date __________ Block _____ For each of the given diagrams, fill in the appropriate formulas for the angle measures and lengths of the segments. Q m QOP P O m QRP R Circle with Center O D C m ACB A B O m ABD Diameter AB and tangent BD R m QUR T U O Q Segment Relationship: S H m JHL J O m HJL L Segment Relationship: K HJ and LJ are tangents 20 U m VUW X Segment Relationship: Y O V W Secants UXV and UYW M R m PMN N O Segment Relationship: P Secant MRN and tangent MP Segment Relationship: B O Arc Relationship: E C D BD OC m FG F O length of FG G 21 Review for quiz WS1 Geometry -- chapter 9 Name _____________________________ Date ______________ Block _________ ________________________________________________________________ ___________________________________________________________________ 22 _____________________________________________________________________ ______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 23 ________________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 24 ___________________________________________________________________ ___________________________________________________________________ _____________________________________________________________________ _____________________________________________________________________ 25 ________________________________________________________________________ ________________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 26 _____________________________________________________________________ _____________________________________________________________________ ______________________________________________________________________ 27 _____________________________________________________________________ ______________________________________________________________________ ______________________________________________________________________ ________________________________________________________________________ 28 ______________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ 29 ______________________________________________________________________ ________________________________________________________________________ 30 _______________________________________________________________________ 31 Review for Quiz WS2 Geometry – chapter 9 Name ____________________________ Date ______________ Block ______ In Circle E, m BD =200, m DF =1800, and m AF =450. GA is tangent to circle E at A. Find the following: 1. m<CAG = _______ 2. m<GCA = _______ 3. m<CGA = _______ _____________________________________________ In the figure, XY is tangent to circle Z at X. 4. If m XW = 950, find m<YXW. _______ 5. If m<YXW = 1000, find m XW . _______ 6. If m XW = x + 15, find m<YXW in terms of x. _______ ________________________________________________________________________ In Circle P, m<LPJ = 300 and m<KMJ = 450. Find the following: 7. m<LMP = _______ 8. m<JPK = _______ 9. m<MJK = _______ 10. m<LPM = _______ 11. m MLJ = _______ 12. m<MPK = _______ 13. m<JPK = _______ 14. m LJ = _______ 15. m KM = _______ 16. m<PLM = _______ 32 In Circle J, 17. SL JP KL at S. _______ For question 18 & 19, give answers in simplest radical form! 18. IF JL = 4, and JS = 1, What is KS? _______ What is KL? _______ 19. IF JK = 26 and JS = 11, What is KS? _______ What is KL? _______ ______________________________________________________________________ Determine the length of an arc with the given central angle measure, m<P, in a circle with radius r. Give answers in terms of and then rounded to the nearest tenth. 20. m<P = 400; r = 6 21. m>P = 200; r = 8 22. m>P = 1180; r = 30 23. m>P = 1300; r = 61 ______________________________________________________________________ Determine the degree measure of an arc with the given length, L, in a circle with radius r. Give answers rounded to the nearest degree. 24. L = 27; r = 5 25. L = 100; r = 79 26. L = 35; r = 11 27. L = 2.3; r = 85 33 Thm: If 2 chords intersect inside a circle the products of the segments on each chord are equal ____________________________________________________________________ 34 Thm: If 2 secants are drawn to a circle from an external point the products of the external secant and the whole secant are equal Proof of theorem: ______________________________________________________________________ 35 Thm: If a secant segment and a tangent segment are drawn to a circle from an external point the product of the secant segment and its external segment is equal to the square of the tangent segment Proof of theorem: _____________________________________________________________________ 36 Segments in Circles WS Geometry Name _____________________ Date __________ Block _____ B C 1. Chords AC and BD intersect at point E. a) If AE = 5, AC = 13, and DE = 10, find BE. A b) If AE = 3, CE = 4x+1, DE = 9, and BE = 2x-1, find x. D c) If AC bisects BD , AE = 8, and EC = 32, find BD. 2. In Circle O, diameter HJ is perpendicular to chord FG at K. If HO = 13 and FG = 10, how far is the chord from the center of the circle? H G O K J F P 3. In Circle O, tangent PT and secant PBA intersect at point P, outside the circle. a) If PT = x, PB = 3, and AB = 13, find x. B T b) If PT = 3, PB = x, and AB = 8, find x. A P 4. Secants PBA and PCD intersect at point P, outside the circle. a) If PB = 8, PA = 18, PC = 9, and PD = x, find x. B C b) If PB = 4, AB = 17, PC = x, and CD = 5, find x. c) If PB = 6, AB = 9, PC = 8, and PD = x, find x. D A 37 Circle Proofs WS 1 Geometry Name ___________________ Date _______ Block _______ 1. Given: Circle O with diameter AOD , tangent CA , mAB 2mAE Prove: ADOA = OCBD 2. Given: Circle O, tangents PR and PV Prove: RPO VPO 3. Given: CT CH Prove: HA TD 4. Given: chords AB and CD of circle O intersect at E, an interior point of circle O; chords AD and CB are drawn. Prove: (AE)(EB) = (CE)(ED) 38 5) 6) 7) 8) ________________________________________________________________________ 9) 39 Circle Proof WS 2 Geometry Name __________________ Date ___________ Block _______ ______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ 40 ______________________________________________________________________ _______________________________________________________________________ _______________________________________________________________________ _______________________________________________ 41 _______________________________________________________________________ _______________________________________________________________________ ______________________________________________ ________________________________________ 42 Review Chapter 9 WS 1 Geometry 1. 2. Name _____________________________ Date ___________ BlockA_________ 3. 8 |----------x------------| y 3 2 x 4 E x B C T D R S 4 10 ________________________________________________________________________ 80 o o 280 4. A 5. 6. 7. 310 o 8. x B x O C O x 120 o 90 o 100 o B x A A B x B A B 70 A D ________________________________________________________________________ C H A 3y 9.o 10. o 11. 12. 100 o 13. B B 100 70 C E O x G 40 o F x x B o y y O 60 O x A o A C O x C O B D ________________________________________________________________________ 80 o x 100 o 80 o 14. 15. 16. 17. 18. 70 A B A o A B x O x C C B 30 o O D O C B A O D D B A 70 o O x C C D D x ________________________________________________________________________ 120 o 19. 20. 21. 22. 23. 4y A 260 o A x D x y E 70 o x H T B T G T ________________________________________________________________________ C C A o 24. 100 o 25. A 50 C 26. 27. 40 o 28. 2y D A 40 o C 100 o x x O T 100 A B A 0 O B D 1400 B O x x B x T y B O O D C A O O y 80 o x O O O x A 2y A D D 43 29. A 5y 30. 60 o 31. 32. 260 o 33. 3y 3y E O O O y y 100 o B O O 2y y y D x x x x x C ________________________________________________________________________ 34. given: AB is tangent; AC is secant; BC, DE, FC B A 4 1 are chords; mEB 50 ; mBC 4x 50 ; E mCD x ; mDF x 25 ; mFE x 15 Find: mBC m 1 O F 3 2 mCD m 2 mDF m 3 mFE m 4 ________________________________________________________________________ C C D 35. 36. 37. 38. D x 8 B E 12 C 4 A 6 E A 16 O C B 2 A B 5 B A 4 O O D Find AB C AB = 8 Radius = ? D PD = 12 ________________________________________________________________________ 39. 40. 41. O O 3 x 50 o O x 60 o x radius = 4 ________________________________________________________________________ A D 42. 43. 44. In circle O, radii OA, OB , G and chord AB are drawn. If OA = 2x+8, OB = x+24, and AB = 3x-8, find OA, AB, and m<AOB. O 60 o C O x B x F E m DFE = 170 o 44 P x Circles Chapter Review WS 2 Geometry Name _______________________ Date __________ Block ______ NOTE: Diagrams may not be to scale!!!!!! 1. Find the arc length for Circle P with radius 6 if m<P = 40 (nearest tenth). 2. Find the measure of the central angle that intercepts an arc of length 27 on a circle with a radius of 5 (nearest degree). __________________________________________ 3. If JP KL at S, JK = 26, and JS = 11, find: J KS = _____ KL = _____ _____________________________________ 4. If m<LPJ = 30 and m<KMJ = 45, find: JK = ______ S K J L P L MK = ______ P K LM = ______ m<LMP = ______ m<PLM = ______ ______________________________________ 5. If mBD 20, mDF 180, mAF 45 , find: M A m<CAG = ______ B C G m<GCA = ______ F E D m<CGA = ______ ________________________________________________________________________ Find the value of x. Show algebra for questions 6 - 25. 11 6. 7. 8. 16 12 x 6 x 4 x 16 4 7 45 9. 10. 6 3 100o 11. 250 o x x x 50o 12. 13. 14. x 145 o x 63 o 15 10 x 15. 16. (nearest tenth) 17. (nearest tenth) 6 x+3 2x x+3 130 o x x 4 70 o 18. 4x 19. 20. S R 87 o 94 o x 54 T m TSR 4 x 15 m RTS 5 x 15 3x+9 46 21. 22. 23. 1 87 o 26 o 3x-9 92o 5x+9 2 x m 1 2.5 x m 2 1.5 x 14 24. P 4x 3x Q R 5x Find m R . ________________________________________________________________________ 26. In Circle O, FA is tangent, FEDB is a secant, ADC and AB are chords, m CE = 40, m AB = 130, and m<CAB = 60. C a) m BC = _____ b) m<EBA = _____ c) m<ADE = _____ B D E F O d) m<F = _____ e) m<FAC = _____ A 47 27. In the accompanying diagram of Circle O with inscribed isosceles triangle ABC, AB AC , m BC = 60, FC is a tangent and secant FBA intersects diameter CD at E. A D a) m<ABC = _____ E b) m AD = _____ O c) m<DEB = _____ d) m<AFC = _____ B C e) m<BCF = _____ F _______________________________________________________________________ 28. In the accompanying diagram of Circle O, secant ABP , secant CDP , and chord AC is drawn; chords AD and BC intersect at E, tangent GCF intersects circle O at C, and m AB : m BD : m DC : m CA = 8:2:5:3. A G a) m CA = ______ b) m<ACB = ______ c) m<P = ______ d) m<AEB = ______ O B C E e) m<DCF = ______ D P F ________________________________________________________________________ 29. In the accompanying diagram of Circle O, AOED is a diameter, PD is a tangent, PBA is a secant, chords BD and BEC are drawn, m<DAB = 43, and m<DEC = 72. a) m<BDP = ______ A B b) m AB = ______ c) m AC = ______ P O d) m<P = ______ E e) m<CBD = _____ D C 48 Chapter 9--Theorems/Corollaries/Postulates Formulas to know: C d A r2 x length of arc: l d ; x = measure of central 360 ________________________________________________________________________ Hints: draw radii to endpts. of a chord [look for special right s] find isosceles s formed w/ radii and a chord find right s formed w/ tangent [radii or diameter a side of the ] _______________________________________________________________________ Basics 1. line tangent to line is to radius @ pt. of tangency 2. [coplanar line & ] line to radius @ pt. on line tangent to 3. tangents to from exterior pt. are 4. [in 1 or s] arcs chords 5. diameter to chord bisects the chord & its intercepted arc [then can use bisect to midpoint to congruent segs] [then to congruent arcs] 6. diameter bisects a chord to the chord at midpoint of chord 7. [in 1 or s] 2 chords equidist. from center chords 8. 2 inscribed angles which intercept the same arc are 9. an angle inscribed in a semicircle is a right angle 10. quad inscribed in opp. angles are supplementary 11. parallel lines which intersect circle intercept arcs _______________________________________________________________________ Angles 1. central = measure of intercepted arc 2. inscribed = 12 measure of intercepted arc 3. formed by tangent & chord has measure = 12 the intercepted arc 4. [notice this does not work for a secant and chord!] formed by 2 chords which inside has measure = 1 2 sum of the 2 intercepted arcs formed by 2 secants measure = 12 formed by 2 tangents difference of the formed by 1 secant & 1 tangent intercepted arcs _______________________________________________________________________ Segments 1. 2 chords inside products of segments formed on each chord are = 2. 2 secant segs to products of external seg & whole secant for each secant seg are = 3. secant & tangent seg to 5. product of ext. seg & whole secant = tan seg 2 49