Geometry hapter 9 ircle Vocabulary rc Length ngle & Segment Theorems with ircles Proofs
hapter 9: ircles Date Due Section Topics ssignment 9.1 9.2 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, 14-20 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], #16 9.4 rcs and hord Relationships Pg. 347 # 1-9, 12, 18, 22 9.5 Inscribed ngles Pg. 354-355 #2-8, 19-21, 9.6 ngles formed by hords, Pg. 359-361 #1-10, 12-24 even Tangents and Secants 9.7 Lengths of Segments in a Pg. 364 (bottom)- 366 #2-8 even, ircle 14-22 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.369-370 (hpt Rev) #1-24 Pg.371 (hpt Test) #1-18 1
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
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
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
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 = 2 360 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
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
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
Thm: line tangent to a circle the line is perpendicular to the radius @ the pt of intersection onverse: line which is perpendicular to the radius @ 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
3. If P = 12 and P = 6, find. Tangents with ircles Internal Tangent Line Eternal Tangent Line Tangent ircles 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. onverse: 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. onverse: if chords are congruent the chords are equidistant from the center 10
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
orollary 1: 2 inscribed angles which intercept the same arc are congruent. 12
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
Inscribed ngles Notes Geometry Name Date lock 14
ngles 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 E(1-10) 6 17
ther ngle Relationships Geometry Name Date lock 18
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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
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
Review for quiz WS1 Geometry -- chapter 9 Name Date lock 22
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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 = 45 0. 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 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 = 6 21. m>p = 20 0 ; r = 8 22. m>p = 118 0 ; r = 30 23. m>p = 130 0 ; 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 eternal point the products of the eternal 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 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
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
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
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ircle Proof WS 2 Geometry Name Date lock 40
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Review hapter 9 WS 1 Name Geometry Date lock 1. 2. 3. 8 ---------------------- 3 2 4 4 y R S D T E 10 80 4. o 280 o 310 o 5. 6. 7. 8. 120 o 100 o 90 o D 70 o 9. 70 o H 10. 11. 12. 100 o 100 o 13. 3y G y y 40 o E F 60 o D 80 o 80 o 100 14. 15. 16. 17. 18. o 70 o 30 o D D 70 o D D D 120 o 19. 20. 21. 22. 23. 260 o 2y 4y 80 o y y 70 o T D T E T G H T T 100 o D 40 o 2y 24. 25. 50 o 26. 27. 28. 100 o y 100 0 40 o D D D 140 0 D 43
29. 60 o 30. 5y 31. 260 o 32. 3y 33. E y 100 o y 2y y F E 1 3 D 4 2 34. given: is tangent; is secant;, DE, F D 35. 36. 37. 38. 4 2 6 8 E E 16 12 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 = 12 39. 40. 41. P 3 50 o 60 o radius = 4 42. 43. 44. 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
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 6-25. 6. 11 7. 8. 16 12 G F 4 E 6 D 16 4 7 45
9. 10. 11. 6 100 o 3 250 o 50 o 12. 13. 14. 145 o 10 15 63 o 15. 16. (nearest tenth) 17. (nearest tenth) 130 o 70 o 2 4 +3 +3 4 6 18. 19. 20. S R 87 o 94 o T 54 m TSR 4 15 m RTS 5 15 3+9 46
21. 22. 23. 1 26 o 87 o 3-9 92 o 5+9 2 m m 1 2.5 2 1.5 14 24. 4 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
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
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 radius @ pt. of tangency 2. [coplanar line & ] line to radius @ 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