How To Understand The Theory Of Ircles

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1 Geometry hapter 9 ircle Vocabulary rc Length ngle & Segment Theorems with ircles Proofs

2 hapter 9: ircles Date Due Section Topics ssignment Written Eercises Definitions Worksheet (pg330 classroom e.all) Pg. 331 #4, 6, 7, 12-15, 17 Pg.337 #1-3 (mied review-bottom of page) Tangents ircumscribed vs. Inscribed ommon Tangent Tangent ircles Pg. 335(bottom)-337 #1-7, 8(not d), 10, even 9.3 rcs (minor and major) entral < s rc ddition Postulate ongruent rcs Length of an rc Pg. 341 (bottom)- 342 # 1-6, 10, [note m< = m<2], # rcs and hord Relationships Pg. 347 # 1-9, 12, 18, Inscribed ngles Pg #2-8, 19-21, 9.6 ngles formed by hords, Pg #1-10, even Tangents and Secants 9.7 Lengths of Segments in a Pg. 364 (bottom)- 366 #2-8 even, ircle even More Proofs Worksheet Proofs Review Study For Test hapter 9 Etra Practice Pg.349 (Self Test 1) #1-6 Pg.367 (Self Test 2) #1-8 Pg (hpt Rev) #1-24 Pg.371 (hpt Test) #1-18 1

3 ircle Introductory Vocabulary Geometry Name Date lock Use appropriate notation to name the following in the given diagram. Write a short eplanation or definition as needed. circle: center: diameter: radius: chord: arc: semicircle: major arc: minor arc: secant: tangent: inscribed polygon: circumscribed polygon: 2

4 p.330 lass Eercises 1. Name three radii of. 2. Name a diameter. 3. onsider RS and RS. Which is a chord and which is a secant? 4. Why is TK not a chord? 5. Name a tangent to. 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 has a length of 20. Radii and are drawn in, forming an angle with the given measure. Find the length of using your knowledge of isosceles and special triangles. a) m< = 90 b) m< = 60 c) m< = 120 3

5 9-3: rcs and entral <'s & 11-6: rc Length n arc is measured in degrees - Its measure is equal in measure of the central angle which intercepts it. rcs are iff. their central <'s are, with angles 0<θ<360. 4

6 The central angles are equal in measure... While the arcs are equal in measure, the arcs are different in length! rc length is related to circumference... = πd or = 2 π r rc length... l = central 360 measure d central measure l = 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

7 entral ngle & rcs Notes Geometry Name Date lock Find the measure of each arc in the diagram. Use the diagram to answer the following: 6

8 rc 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. Etra Practice: p.341 E(1-13) 7

9 Thm: line tangent to a circle the line is perpendicular to the the pt of intersection onverse: line which is perpendicular to the a point on the circle the line is tangent to the circle * the circle & line must be oplanar! Thm: parallel lines/chords in a circle intercepted arcs are congruent Thm: Tangent segments from an eternal 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

10 3. If P = 12 and P = 6, find. Tangents with ircles Internal Tangent Line Eternal Tangent Line Tangent ircles 9

11 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. onverse: if diameter bisects a chord it is perpendicular to the its midpoint Thm: In the same circle or congruent circles, If 2 chords are equidistant from the center the chords are congruent. onverse: if chords are congruent the chords are equidistant from the center 10

12 lasswork: 1) p.349 ST1 (1-6) 2) p.346 lass Eercises (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 * n angle inscribed in a semicircle right angle Proof of theorem on net page 11

13 orollary 1: 2 inscribed angles which intercept the same arc are congruent. 12

14 orollary 2: n angle inscribed in a semicircle is a right angle. orollary 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 Eamples (p.353 #4 9) Find the value for and y in each question. 13

15 Inscribed ngles Notes Geometry Name Date lock 14

16 ngles formed by a tangent and a chord Notes 15

17 Thm: If 2 chords intersect inside a circle the products of the segments on each chord are equal Proof of theorem: 16

18 Thm: an angle formed by 2 secants/tangents its measure = half the difference of the measures of the intercepted arcs p.359 E(1-10) 6 17

19 ther ngle Relationships Geometry Name Date lock 18

20 19

21 ircle Formulas Geometry Name Date lock For each of the given diagrams, fill in the appropriate formulas for the angle measures and lengths of the segments. Q m QP P R m QRP ircle with enter D m m D Diameter and tangent D T R m QUR U Q Segment Relationship: S H J m JHL m HJL K L Segment Relationship: HJ and LJ are tangents 20

22 U m VUW X Y Segment Relationship: V W Secants UXV and UYW R M m PMN N P Segment Relationship: Secant MRN and tangent MP Segment Relationship: E rc Relationship: D D F m FG length of FG G 21

23 Review for quiz WS1 Geometry -- chapter 9 Name Date lock 22

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33 Review for Quiz WS2 Geometry chapter 9 Name Date lock In ircle E, m D =20 0, m DF =180 0, and m F =45 0. G is tangent to circle E at. Find the following: 1. m<g = 2. m<g = 3. m<g = In the figure, XY is tangent to circle Z at X. 4. If m XW = 95 0, find m<yxw. 5. If m<yxw = 100 0, find m XW. 6. If m XW = + 15, find m<yxw in terms of. In ircle P, m<lpj = 30 0 and m<kmj = 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

34 In ircle J, JP 17. SL 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 = 40 0 ; r = m>p = 20 0 ; r = m>p = ; r = m>p = ; 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 = L = 100; r = L = 35; r = L = 2.3; r = 85 33

35 Thm: If 2 chords intersect inside a circle the products of the segments on each chord are equal 34

36 Thm: If 2 secants are drawn to a circle from an eternal point the products of the eternal secant and the whole secant are equal Proof of theorem: 35

37 Thm: If a secant segment and a tangent segment are drawn to a circle from an eternal point the product of the secant segment and its eternal segment is equal to the square of the tangent segment Proof of theorem: 36

38 Segments in ircles WS Geometry Name Date lock 1. hords and D intersect at point E. a) If E = 5, = 13, and DE = 10, find E. b) If E = 3, E = 4+1, DE = 9, and E = 2-1, find. c) If bisects D, E = 8, and E = 32, find D. D 2. In ircle, diameter HJ is perpendicular to chord FG at K. If H = 13 and FG = 10, how far is the chord from the center of the circle? H K G P F J 3. In ircle, tangent PT and secant P intersect at point P, outside the circle. a) If PT =, P = 3, and = 13, find. b) If PT = 3, P =, and = 8, find. T P 4. Secants P and PD intersect at point P, outside the circle. a) If P = 8, P = 18, P = 9, and PD =, find. b) If P = 4, = 17, P =, and D = 5, find. c) If P = 6, = 9, P = 8, and PD =, find. D 37

39 ircle Proofs WS 1 Geometry Name Date lock 1. Given: ircle with diameter D, tangent, m 2mE Prove: D = D 2. Given: ircle, tangents PR and PV Prove: RP VP 3. Given: T H Prove: H TD 4. Given: chords and D of circle intersect at E, an interior point of circle ; chords D and are drawn. Prove: (E)(E) = (E)(ED) 38

40 5) 6) 7) 8) 9) 39

41 ircle Proof WS 2 Geometry Name Date lock 40

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44 Review hapter 9 WS 1 Name Geometry Date lock y R S D T E o 280 o 310 o o 100 o 90 o D 70 o o H o 100 o 13. 3y G y y 40 o E F 60 o D 80 o 80 o o 70 o 30 o D D 70 o D D D 120 o o 2y 4y 80 o y y 70 o T D T E T G H T T 100 o D 40 o 2y o o y o D D D D 43

45 o 30. 5y o 32. 3y 33. E y 100 o y 2y y F E 1 3 D given: is tangent; is secant;, DE, F D E E D Find are chords; me 50 ; m 4 50 ; md ; mdf 25 ; mfe 15 Find: m m 1 md m 2 mdf m 3 D mfe m 4 = 8 Radius =? D 5 3y y 4 PD = P 3 50 o 60 o radius = In circle, radii,, and chord are drawn. If = 2+8, = +24, and = 3-8, find,, and m<. 60 o G D E m DFE = 170 o F 44

46 ircles hapter Review WS 2 Geometry Name Date lock NTE: Diagrams may not be to scale!!!!!! 1. Find the arc length for ircle 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: KS = KL = 4. If m<lpj = 30 and m<kmj = 45, find: JK = L J K J S P L MK = LM = P K m<lmp = m<plm = 5. If md 20, mdf 180, mf 45, find: M m<g = m<g = m<g = Find the value of. Show algebra for questions G F 4 E 6 D

47 o o 50 o o o (nearest tenth) 17. (nearest tenth) 130 o 70 o S R 87 o 94 o T 54 m TSR 4 15 m RTS

48 o 87 o o m m P Q 3 R 5 Find m R. 26. In ircle, F is tangent, FED is a secant, D and are chords, m E = 40, m = 130, and m< = 60. a) m = b) m<e = c) m<de = D E F d) m<f = e) m<f = 47

49 27. In the accompanying diagram of ircle with inscribed isosceles triangle,, m = 60, F is a tangent and secant F intersects diameter D at E. D a) m< = b) m D = c) m<de = E d) m<f = e) m<f = 28. In the accompanying diagram of ircle, secant P, secant DP, and chord is drawn; chords D and intersect at E, tangent GF intersects circle at, and m : m D : m D : m = 8:2:5:3. a) m = b) m< = c) m<p = d) m<e = e) m<df = G E D F P 29. In the accompanying diagram of ircle, ED is a diameter, PD is a tangent, P is a secant, chords D and E are drawn, m<d = 43, and m<de = 72. F a) m<dp = b) m = P c) m = d) m<p = e) m<d = E D 48

50 hapter 9--Theorems/orollaries/Postulates Formulas to know: 2 r d length of arc: l d ; = 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 ] asics 1. line tangent to line is to pt. of tangency 2. [coplanar line & ] line to pt. on line tangent to 3. tangents to from eterior 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 ngles 1. central = measure of intercepted arc 2. inscribed = 1 2 measure of intercepted arc 3. formed by tangent & chord has measure = 1 2 the intercepted arc [notice this does not work for a secant and chord!] 4. formed by 2 chords which inside has measure = 1 2 sum of the 2 intercepted arcs 5. formed by 2 secants measure = 1 2 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 eternal seg & whole secant for each secant seg are = 3. secant & tangent seg to product of et. seg & whole secant = 2 tan seg 49

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