Circles.notebook
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October19,2012
Aug2510:52AM
KEY STANDARDS ADDRESSED:MM2G3. Students will understand the properties of circles.
a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity.
b. Understand and use properties of central, inscribed, and related angles. c. Use the properties of circles to solve problems involving the length of an arc and the area of a sector.
d. Justify measurements and relationships in circles using geometric and algebraic properties.
Circles.notebook
2
October19,2012
Aug259:25AM
Unit3keyvocabulary
*rateyourknowledgeofeachterm:1. Noclue!
2. Hearditbefore,butitisfuzzy.3.Iknowthistermwell!
radius areaofcirclechord diametertangent spheresecant surfaceareasector volumeofspherescentralangle areaofasectorinscribedangle arclength
MM2G3
Circles.notebook
3
October19,2012
Aug259:31AM
Circles
Whatarethebasicpartsofacircle?
Whatisacircle?
Whatarethepropertiesofchords,tangentsandsecantsincircles?
Whatarethedifferenttypesofangles(andtheirproperties)formedbychords,tangentsandsecants?
lessonpowerpoints
Unit3EssentialQuestions
Spheres
Howdoyoucalculatesurfaceareaandvolumeofasphere?
Howisthesurfaceareaandvolumeofaspherealteredwhentheradiusischanged?
clicktogotospheres
CirclearclengthandsectorsHowcanyouusepropertiesofcirclestosolveproblemsinvolvingthelengthofanarcandtheareaofasector?
lessononarclength
lessononareaofsectors
MM2G3
Circles.notebook
4
October19,2012
Mar3011:40PM
CIRCLES
specialsegments
day1
arcsandchords
day1
tangentsandcircles
day1
circletermsandparts
day1
MM2G3
Circles.notebook
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October19,2012
Apr2910:54AM
classworkforeachtopic
circletermsandparts tangentsandcircles
arcsandchords specialsegments
MM2G3
Chapter 8: Circles name ________________________ Lesson 8-1: Terminology date ______________ Classwork period _____ Sketch Define
1. circle ____________________________________________________ ___________________________________________________________ 2. radius ___________________________________________________ ___________________________________________________________ 3. chord ____________________________________________________ ___________________________________________________________ 4. diameter __________________________________________________ ___________________________________________________________ 5. secant ____________________________________________________ ___________________________________________________________ 6. tangent ___________________________________________________ ___________________________________________________________ 7. point of tangency ___________________________________________ ___________________________________________________________ 8. common tangent ___________________________________________ ___________________________________________________________ 9. congruent circles ___________________________________________ ___________________________________________________________ 10. concentric circles __________________________________________ ___________________________________________________________
11. inscribed ________________________________________________ ___________________________________________________________ 12. circumscribed ____________________________________________ ___________________________________________________________ 13. arc _____________________________________________________ ___________________________________________________________ 14. minor arc ________________________________________________ ___________________________________________________________ 15. semicircle ________________________________________________ ___________________________________________________________ 16. major arc ________________________________________________ ___________________________________________________________ 17. arc length ________________________________________________ ___________________________________________________________ 18. circumference ____________________________________________ ___________________________________________________________ 19. sphere __________________________________________________ ___________________________________________________________ 20. great circle _______________________________________________ ___________________________________________________________
SMART Notebook
x
12
BO
A
C
12
7CB
A
x 9
x
12
1616
x
918
Chapter 8: Circles Name _____________________________ Lesson 8-3: Tangents Date ______________ Classwork Period ___ Find x. Assume that segments that appear to be tangent are tangent lines. Round answers to the nearest tenth. 1. x = ____________ 2. x = ____________ 3. x = ____________ 4. x = ____________
5. x = ____________
10
4
3
O
F
AB
C
D E
24
O
E
A
B
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C6. Assume points A, E, and D are tangent to circle O. Find BC . 7. Assume D, E and F are tangent to circle O. Find AC . If BD bisects AC , BD AC , AB =13, and AC = 24. Find the indicated values.
8. BE = _______ 9. DE = _______ 10. If AB = 12 and m A =30, find BE = _________ and AE = __________. 11. If BE = 5 and m B = 60, find AB = ________ and AE = ___________.
E
B
D
A C
SMART Notebook
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Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Classwork Period _________ 1. 2. 3.
AC BD AC BD AC BD m 94ABC = m 4AE = m 12AC =
Find AB ______ Find AC ______ m 8DE = Find the radius______ 4. 5. 6.
Find AB ______ GB GE GB GE Find ABF ______ m 10EF = m 5EF = Find ABD ______ Find DF _______ Find mCA = ____
7. 8.
AC BD 17DA = AC DF
m 8ED = 100mAC = Find AC ______ Find mDF _____
9. Suppose a chord is 9 meters from the center of a circle. It is 20 meters long. Find the length of the radius. ___________
10. Find the length of a chord 4 inches from the center of a circle with a radius of 5 inches. __________
SMART Notebook
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B
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C
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O
A
B
D
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Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Classwork Period ___ Secants, chords and tangents are shown. For questions 1 - 6, refer to the figure below and find the indicated value. 1. If CE = 3, DE = 6, and AE = 2, find BE. 2. If AE = 3, BE = 5, and DE = 2, find CE. 3. If AE = 3, BE =6 , and CE = 4, find DE. 4. AE = 12, BE = 18, and DE = 9, find CE. 5. If AE = 3.4, BE = 5.2, and CE = 2, find DE. 6. If AE = 2x, BE = 4x, CE = 8, and DE = 16, find x. For questions 7 - 11, refer to the figure below and find the indicated value. 7. If BC = 3 and BD = 12, find AB. 8. If AB = 6 and BD = 12, find BC. 9. If BC = 4 and CD = 12, find AB. 10. If AB = 6 and BD = 9, find BC. 11. If AB = 10 and BC = 5, find CD. For questions 12 - 21, refer to the figure below and find the indicated value. 12. If AC = 12, BC = 4 and CE = 8, find CD. 13. If CE = 9, CD = 4, and BC = 3, find AB. 14. If DE = 3, DC = 9 and BC = 6, find AB. 15. If AB = 17, BC = 3, And CD = 6, find CE. 16. If DE = 8, CD =7, and AC = 21, find BC. 17. If CE =15, DE = 10, BC = 4, find AB. 18. If CD = 8, DE = 10, and AB = 10, find BC. 19. If BC = 5, AB=7, CD = x and DE = 5x, find x. 20. If BC = 12, AB = 13, CD = x, and DE = 2x, find x.
SMART Notebook
Circles.notebook
6
October19,2012
Apr2912:47PM
homeworkforeachtopic
circletermsandparts tangentsandcircles
arcsandchords specialsegments
MM2G3
2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 25
Discovering and Proving Circle Properties
C H A P T E R
6Content SummaryIn Chapter 6, students continue to build their understanding of geometry as theyexplore properties of circles. Some of these properties are associated with linesegments related to circles; other properties are associated with arcs and angles.A circle is defined as a set of points equidistant from a fixed point, its center.
Line Segments Related to CirclesThe best-known line segments related to a circle are its radius and diameter.Actually, the word radius can refer either to a line segment between a point on the circle and the center, or to the length of such a line segment. Similarly,diameter means either a line segment that has endpoints on the circle andpasses through the center, or the length of such a line segment.
The diameter is a special case because it is the longest chord of a circle; a chordis a line segment whose endpoints are on the circle. Another line segmentassociated with circles is a tangent segment, which touches the circle at just onepoint and lies on a tangent line, which also touches the circle at just one pointand is perpendicular to the radius at this point. Students learned about thesesegments in Chapter 1, and Lesson 6.1 provides a quick review.
Arcs and AnglesA piece of the circle itself is an arc. If you join each endpoint of an arc to the centerof the circle, you form the central angle that intercepts the arc. The size of the arc can be expressed in degreesthe number of degrees in the arcs central angle. Thischapter explores several such relationships among arcs, angles, and segments.Students also write paragraph and flowchart proofs to confirm the universality ofthese relationships.
The size of the arc can also be expressed in length. The arc length is calculated byusing the total circumference, or the distance around the circle. The number isdefined to be the circumference of any circle divided by that circles diameter; or, thecircumference is times the diameter. For example, if an arc is 14 of the completecircle, then its central angle measures 14 of 360, and its length is
14 of the circles
circumference.
Summary ProblemDraw a diagram of a central angle intercepting a chord of a circle and its arc,as shown in the picture.
Move points A, B, and C to different locations to illustrate the ideas of the chapter.
Questions you might ask in your role as student to your student:
What concepts are illustrated in the original drawing?
How could you move each of the points A, B, and C to show an inscribedangle?
How could you move each of the points A, B, and C to show tangent segments?
A
C
B
Radius
Diameter
Chord
Tangent
(continued)
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26 Discovering Geometry: A Guide for Parents 2008 Key Curriculum Press
Chapter 6 Discovering and Proving Circle Properties (continued)
How could you move points A, B, and C to show an angle inscribed in asemicircle?
How could you move points A, B, and C to show parallel lines interceptingcongruent arcs?
Sample Answers The original drawing shows a central angle, a sector, a chord, and an arc. If thecenter, C, is moved to lie on the circle, an inscribed angle is formed.
If C is moved to be outside the circle, and A and B are moved to make AC and BCtangents, then those segments are congruent. Or, the chord in the original diagramcould be rotated at one of its ends until it becomes a tangent segment.
If C is moved until AC is a diameter and B remains on the circle, ABC is a rightangle inscribed in a circle.
You would need to add another point and put C on the circle to show two chords.Parallel chords intercept equal arcs if they are equidistant from the center.
Of course many other answers are possible. Encourage your student to think ofmultiple ways that A, B, and C could be moved to illustrate these same concepts.
AC
B
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Chapter 6 Review Exercises
Name Period Date
1. (Lesson 6.1) Given tangent AB, find mOAB, mAOB,and mABO.
2. (Lessons 6.2, 6.3) Find the unknown measures or lengths.
3. (Lesson 6.3) ABC is an equilateral triangle. Find mAB.
4. (Lesson 6.4) Write a paragraph proof to prove the following:
Given: Circle A with diameters EC and BD.
Prove: ED BC
5. (Lessons 6.5, 6.7) Given that the circumference of circle A is 24 in., find the radius of the circle and the length of BDC.
6. (Lessons 6.1, 6.3) AB and BC are tangents to the circle as shown.AC ED. Find a and b.
C
B
D
E
A
15 m
a
b100
140
C
A
B120
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B
C
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AE
A
BC
a
c d
b10 cm
130
A B
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O
310
2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 27
DG4GP_905_06.qxd 12/27/06 10:23 AM Page 27
4. Diameters CE and BD on circle A intersect to formcongruent vertical angles, so mBAC mDAE.The measure of an arc equals the measure of itscentral angle. Therefore, mBC mED because themeasures of their central angles are equal.
5. C 2r 24 in.; therefore, r 12 in.
mBDC 360 120 240
length of BDC 2346
00 (24) 16 in.
6. mCD mAE a Parallel secants inter-cept congruent arcs.
100 a a 140 360 360 in a circle.
a 60 Solve.
b 15 m Tangent segments fromthe same point arecongruent.
1. mAOB 90 Tangent is perpendi-cular to radius.
mAC 360 310 50 360 in a circle.
mAOC mAC 50 Central angle isequal to arcmeasure.
50 90 mABO 180 Triangle sum.
mABO 40 Subtraction.
2. b 90 Diameter is perpen-dicular to the chord.
d 10 cm Diameter perpendi-cular to a chordbisects the chord.
c 180 130 50 180 in a semicircle.
a 12(130) Inscribed angles.
ma 65 Division.
3. AB BC AC Equilateral triangle.
mAB mBC mAC Congruent arcs ofcongruent chords.
mAB mBC mAC 360 360 in a circle.
mAB mAB mAB 360 Substitution.
3mAB 360 Combine like terms.
mAB 120 Division.
28 Discovering Geometry: A Guide for Parents 2008 Key Curriculum Press
S O L U T I O N S T O C H A P T E R 6 R E V I E W E X E R C I S E S
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SMART Notebook
D
HEB
A
C
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O
CA
Chapter 8: Circles Name____________________ Lesson 8-3: Tangents Date ______________ Homework Period ___ For questions 1 - 3, O and R are centers of circles. Find the indicated value. 1. 2. 3.
OR = ________ AB = _______ AD = _____ BC = _____ Refer to the accompanying figure for questions 4 - 6. Find the indicated values. 4. If HD = 12 and EH = 9, DE = _____________. 5. If DE = 17 and BH = 9, CD = _____________. Refer to the accompanying figure for questions 7 - 10. * is tangent to circle O. 6. If AC = 4 and OC = 3, then AO = ___________. 7. If OC = 15 and AC = 20, then AO = _______________. 8. If m OAC =30 and AO = 10, then OC = ___________. 9. If m OAC = 60 and OC = 4 3 , then AC = _________.
62
O R 5OB1
RA1
43
109 BA
CD
O
DC
A
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CD and BC are tangent to circle O. Refer to the accompanying figure and find the indicated values. 10. If OC = 20 and OD = 12, then BC = ___________. 11. If OC = 4 2 and CD = 4, then OD = ____________. 12. If AD = 10 and CD = 12, then OC = _____________. 13. If OC = 5 3 and CD = 5 2 , then AD = ____________. 14. If m OCD = 30 and OD = 6, then OC = ____________ and CD = ___________. 15. If m COD = 60 and CD = 4 3 , then OC = ___________ and AD = ____________.
SMART Notebook
E
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Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Homework Period _________ 1. 2. 3.
AE EC AC BD AC BD Find m AEB ______ m 10AC = m 22ED = Find m AE ______ DC = 32 Find m EB _____ 4. 5. 6. AC DF AC DF mGE = 7 80mAF = m BG = 4 GF = 25 60mCD = Find mGE _______ Find m DF ______ Find mAC _______ 7. 8. Suppose that a circle has a radius of 35 units and a chord
is 56 units. Find the distance from the center to the chord. __________.
210mBE = Find mCD _____ 9. Suppose the diameter of a circle is 20 feet long and a
non-intersecting chord is 12 feet long. Find the distance between the chord and the center. __________
SMART Notebook
B
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B
O
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A C
B
OD
A
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Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Homework Period ___ For questions 1 - 6, refer to the figure below and find the indicated value. 1. If AB = 25, BC = 3, and BE = 15, find BD. 2. If AB = 4, BC = 9, and BD = 6, find BE. 3. If AC = 16, AB = 4, and BE = 8, find DE. 4. If DE = 17, BD = 7, and AB = 5, find AC. 5. If AB = 3, BC = 5 and BE = 8, find BD. 6. If BE = 16, BD = 4, and B is the midpoint of AC, find AB. In the accompanying diagram, * is tangent to circle O at D and * is a secant. 7. If AD = 9 and AB = 3, find AC. 8. If BC = 15 and AB = 1, find AD. 9. If AD = 8 and AB = 4, find AC. 10. If AB = 4 and BC = 5, find AD. 11. If AD = 3 5 and AB = 3, find BC. In the accompanying diagram, two secants are drawn from the same point. 12. If AB = 5, AC = 8, and AD = 2, find DE. 13. If AB = 3, BC = 7 and AE = 15, find AD. 14. If AB = 6, BC = 12, and AD = 4, find DE. 15. If AC = 20, AD = 8, and DE = 2, find AB. 16. If AB = 5, AD = 8 and DE = 2, find BC. 17. If B is the midpoint of AC , and AD = 8, and DE = 17, find AC.
SMART Notebook
Circles.notebook
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October19,2012
Nov910:02AM
CentralandInscribedAngles
Acentralangleofacircleisananglewhosevertexisthecenterofthecircle.
CentralAngle
Aninscribedangle isanangleinacircle,whosevertexisonthecircleandwhosesidescontainchordsofthecircle.
InscribedAngle
A
B
D InterceptedArc
isaninscribedangle.istheinterceptedarc.
MM2G3
A
B
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B
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EC
A
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A
B
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40D
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Examples
A
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C24040
270
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100200 E
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125
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Circles.notebook
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Nov118:54AM
A
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Circles.notebook
10
October19,2012
Apr309:18AM
Anarcispartofacircle'scircumference
ArcsinCircles
IncircleO,theradiusis8,andthemeasureofminorarcABis110degrees.FindthelengthofminorarcABtothenearestinteger.
Homework
http://www.regentsprep.org/Regents/math/geometry/GP15/PcirclesN4.htmOn line practice
MM2G3
Circles.notebook
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October19,2012
Apr309:06AM
AreaofaCircleMM2G3
Circles.notebook
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Sep152:14PM
AreaofaSector
Whatistheareaofasemicircle?r2
Whatistheareaofaquartercircle?r2
Whatistheareaofanysectionofacircle?r2
Whatifwearenotgiventheangle?r2
Findtheareaofasectorwiththecentralangleof60andaradiusof10.Expresstheanswertothenearesttenth.
A=r2
A=(10)2A=52.4
Findtheareaofasectorwithanarclengthof40cmandaradiusof12cm.
A=(12)2
A=240sq.cm
MM2G3
Circles.notebook
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October19,2012
Nov309:01AM
A
C
D
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A
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AreaofaSector
ArcLength/Measure
Circles.notebook
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October19,2012
Sep153:06PM
Segment of a CircleAsegmentofacircleistheregionboundedbyachordandthearc.
Segment
FindingtheareaofasegmentofacircleFirst,youmustfindtheareaofathesectorofthecircle
Second,findtheareaofthetriangle
Last,subtracttheareaofthetrianglefromtheareaofthesectortofindthesegmentofthecircle
Inotherwords:Asegment=Asector Atriangle
MM2G3
Circles.notebook
15
October19,2012
Sep153:14PM
Findtheareaofasegmentofacirclewithacentralangleof120degreesandaradiusof8.Expressanswertonearestinteger.
Startbyfindingtheareaofthesector
A=(8)2
A= (64)A=67.02
Now,findtheareaofthetriangle.Droppingthealtitudeformsa306090degreetriangle.Usingtrig.(orthe306090rules),findthealtitude,whichis4,andtheotherleg,whichis43.
A=bh
A= (43)(4)A=13.856
Wehavetwotriangles,sowehavetomultiplythatby2.
A=27.71
Asegment=Asector Atriangle
A segment=67.0227.71
Asegment=39.3
MM2G3
Circles.notebook
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October19,2012
Mar3011:13PM
TermsanddefinitionsReview:Acircleisthesetofallpointsinaplanethatareequidistant(thelengthoftheradius)fromagivenpoint,thecenter,ofthecircle.
Achordisasegmentontheinteriorofacirclewhoseendpointsareonthecircle.
Adiameterisasegmentbetweentwopointsonacircle,whichpassesthroughthecenterofthecircle.
Anarcisaconnectedsectionofthecircumferenceofacircle.Anarchasalinearmeasurement,whichistheportionofthecircumference,andanarchasadegreemeasurement,whichisaportionofthe360degreecircle.
Ifacircleisdividedintotwounequalarcs,theshorterarciscalledtheminorarcandthelongerarciscalledthemajorarc.
Ifacircleisdividedintotwoequalarcs,eacharciscalledasemicircle.
Drawacircleandlabelthepartslistedabove
MM2G3
Circles.notebook
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October19,2012
Mar3011:21PM
Asecantline isalinethatintersectsacircleattwopointsonthecircle.
Atangentline isalinethatintersectsthecircleatexactlyonepoint.
Acentralangleofacircleisananglewhosevertexisthecenterofthecircle.
Aninscribedangleisanangleinacircle,whosevertexisonthecircleandwhosesidescontainchordsofthecircle.
Asectorofacircleisaregionintheinteriorofthecircleboundedbytworadiiandan
MM2G4
Circles.notebook
18
October19,2012
May122:52PM
Teacher'stestpage:clickonthelinktoopendifferentversionsoftestsforunit3theseweremadeusingthemcdougallitteltestgenerator
Reviewitems
version1 version2 part2
review#1 Review#2
MM2G3
SMART Notebook
SMART Notebook
SMART Notebook
SMART Notebook
SMART Notebook
Circles.notebook
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October19,2012
Mar3011:30PM
http://teachers.henrico.k12.va.us/math/igo/08Circles/8_1.html
http://www.ies.co.jp/math/java/geo/circles.html
http://resource.sbo.accomack.k12.va.us/itrt/Geometry/Geometry.htm
https://www.georgiastandards.org/Frameworks/GSO%20Frameworks/912%20Accelerated%20Math%20I%20Student%20Edition%20Unit%203%20Circles%20and%20Spheres.pdf
helpfulwebsites
http://www.glencoe.com/sec/math/studytools/cgibin/msgQuiz.php4?isbn=0078884845&chapter=10&title=ct&&headerFile=X
interactivepracticetest
MM2G3
http://teachers.henrico.k12.va.us/math/igo/08circles/8_1.htmlhttp://www.ies.co.jp/math/java/geo/circles.htmlhttp://resource.sbo.accomack.k12.va.us/itrt/geometry/geometry.htmhttps://www.georgiastandards.org/frameworks/gso%20frameworks/9-12%20accelerated%20math%20i%20student%20edition%20unit%203%20circles%20and%20spheres.pdfhttp://www.glencoe.com/sec/math/studytools/cgi-bin/msgquiz.php4?isbn=0-07-888484-5&chapter=10&title=ct&&headerfile=x
Circles.notebook
20
October19,2012
Apr309:23AM
CircleOwithtangent.
http://www.regentsprep.org/Regents/math/geometry/GP15/PracBig.htmanswers
MM2G3
http://www.regentsprep.org/regents/math/geometry/gp15/pracbig.htm
Circles.notebook
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October19,2012
Apr309:20AM
CircleOwithtangentMN
http://www.regentsprep.org/Regents/math/geometry/GP15/PracBig.htm
answers
MM2G3
http://www.regentsprep.org/regents/math/geometry/gp15/pracbig.htm
Circles.notebook
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WATERWHEEL Acircularwaterwheelisdividedinto10evenpartsbythespokes.Iftheradiusofoneofthespokesis5feet,whatistheareaofoneofthesections?
MM2G3
Circles.notebook
23
October19,2012
May510:08PM
CyclicQuadrilateralsAcyclicquadrilateralisafoursidedfigureinacircle,witheachvertex(corner)ofthequadrilateraltouchingthecircumferenceofthecircle.Theoppositeanglesofsuchaquadrilateraladdupto180degrees.
InthecircleObelow,whatarethemeasuresofthenumberedangles?
http://www.regentsprep.org/Regents/mathb/5A1/CircleAngles.htmangleswithinacircle
http://www.quia.com/quiz/797276.html?AP_rand=640693805
quizoncircles
http://www.regentsprep.org/Regents/math/geometry/GP14/CircleSegments.htmsegmentsinacircle
MM2G3
http://www.regentsprep.org/regents/mathb/5a1/circleangles.htmhttp://www.quia.com/quiz/797276.html?ap_rand=640693805http://www.regentsprep.org/regents/math/geometry/gp14/circlesegments.htm
Circles.notebook
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October19,2012
Apr309:10AM
Findtheareaofasectorwithacentralangleof60degreesandaradiusof10.Expressanswertothenearesttenth.
EXAMPLE
MM2G3
Circles.notebook
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Apr309:23AM
MM2G3
Circles.notebook
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October19,2012
Jul2111:42AM
KEY STANDARDS ADDRESSED:
MM2G4. Students will find and compare the measures of spheres.
a. Use and apply surface area and volume of a sphere.
b. Determine the effect on surface area and volume of changing the radius or diameter of a sphere.
Circles.notebook
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October19,2012
Apr2912:44PM
powerpointonchangingradius
Spheresthebasics
SpheresHW Spheresclasswork
On line practice with spheres
MM2G4
Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Homework period ___
Find the surface area of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .
1) radius is 12 cm
2) diameter is 16 m
3) diameter is 1 ft
4) radius is 9 in
5) radius is 6 mm
Find the volume of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .
1) radius is 1 m
2) diameter is 18 ft
3) diameter is 16 cm
4) radius is 12 in
5) radius is 8 mm
SMART Notebook
Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Classwork period ___
Fill in the chart using the given information about spheres. Write your answers in terms of .
(256/3)
72
288
100
2
6
1/8
VolumeSurface AreaDiameterRadius
SMART Notebook
Attachments
circleterms.ppt
circlepropertiesandHW.pdf
tangents.ppt
angleformulas.ppt
circletangentsandtheorems.ppt
arcsandchords.ppt
specialsegments.ppt
spheres.ppt
angleformulasHW.pdf
arcsandchordsclasswork.pdf
arcsandchordsHW.pdf
circlepartsClasswork.pdf
cirlcepartsHW.pdf
specialsegmentsclasswork.pdf
specialsegmentsHW.pdf
spheresclasswork.pdf
spheresHW.pdf
tangentsclasswork.pdf
tangentsHW.pdf
areasofsectorsandsegmentsHW.pdf
AreaSectorSegment912quiz.pdf
circles+test.tst
unit3testcircles.tst
unit3part2test.tst
circlesreviewsheet.tst
unit3part2review.tst
MA1G5bspheres.ppt
unit3overviewpage.pdf
AreaofaSector.ppt
practiceonarclengthandareaofsectors.pdf
Lesson 8-1
Circle Terminology
Lesson 8-1: Circle Terminology
72.psd
Circle Definition
Circle :
The set of points coplanar points equidistant from a given point.
The given point is called the CENTER of the circle.
The distance from the center to the circle is called the RADIUS.
Center
Radius
Lesson 8-1: Circle Terminology
Definitions
Chord :
The segment whose endpoints lie on the circle.
Chord
Diameter :
A chord that contains the center of the circle.
Diameter
Secant :
A line that contains a chord.
Secant
Tangent :
A line in the plane of the circle that intersects the circle in exactly one point.
Point of Tangency :
The point where the tangent line intersects the circle.
Tangent
Lesson 8-1: Circle Terminology
Example: In the following figure identify the chords, radii, and diameters.
Chords:
Radii:
Diameter:
Lesson 8-1: Circle Terminology
Circles that have congruent radii.
2
2
Circles that lie in the same plane and have the same center.
Definitions
Concentric circles :
Congruent Circles :
Lesson 8-1: Circle Terminology
Polygons
A polygon inside the circle whose vertices lie on the circle.
Inscribed Polygon:
Circumscribed Polygon :
A polygon whose sides are tangent to a circle.
Lesson 8-1: Circle Terminology
ARCS
The part or portion on the circle from some point B to C is called an arc.
Arcs :
Semicircle:
An arc that is equal to 180.
Example:
A
B
C
Example:
Lesson 8-1: Circle Terminology
Minor Arc & Major Arc
Minor Arc :
A minor arc is an arc that is less than 180
A minor arc is named using its endpoints with an arc above.
A
B
Example:
Major Arc:
A major arc is an arc that is greater than 180.
A major arc is named using its endpoints along with another point on the arc (in order).
A
B
C
Example:
Lesson 8-1: Circle Terminology
Example: ARCS
Identify a minor arc, a major arc, and a semicircle, given that is a diameter.
Minor Arc:
Major Arc:
Semicircle:
Lesson 8-1: Circle Terminology
CD
FB
,,,
CEFEDCDFEFCD
BC
ABC
ABC
,,
ABBFCE
,,,
,,
OBOFOD
OEOCOA
,,,
DEECCFDF
,,,
CEDCFDEDFECF
AB
SMART Notebook
2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 25
Discovering and Proving Circle Properties
C H A P T E R
6Content SummaryIn Chapter 6, students continue to build their understanding of geometry as theyexplore properties of circles. Some of these properties are associated with linesegments related to circles; other properties are associated with arcs and angles.A circle is defined as a set of points equidistant from a fixed point, its center.
Line Segments Related to CirclesThe best-known line segments related to a circle are its radius and diameter.Actually, the word radius can refer either to a line segment between a point on the circle and the center, or to the length of such a line segment. Similarly,diameter means either a line segment that has endpoints on the circle andpasses through the center, or the length of such a line segment.
The diameter is a special case because it is the longest chord of a circle; a chordis a line segment whose endpoints are on the circle. Another line segmentassociated with circles is a tangent segment, which touches the circle at just onepoint and lies on a tangent line, which also touches the circle at just one pointand is perpendicular to the radius at this point. Students learned about thesesegments in Chapter 1, and Lesson 6.1 provides a quick review.
Arcs and AnglesA piece of the circle itself is an arc. If you join each endpoint of an arc to the centerof the circle, you form the central angle that intercepts the arc. The size of the arc can be expressed in degreesthe number of degrees in the arcs central angle. Thischapter explores several such relationships among arcs, angles, and segments.Students also write paragraph and flowchart proofs to confirm the universality ofthese relationships.
The size of the arc can also be expressed in length. The arc length is calculated byusing the total circumference, or the distance around the circle. The number isdefined to be the circumference of any circle divided by that circles diameter; or, thecircumference is times the diameter. For example, if an arc is 14 of the completecircle, then its central angle measures 14 of 360, and its length is
14 of the circles
circumference.
Summary ProblemDraw a diagram of a central angle intercepting a chord of a circle and its arc,as shown in the picture.
Move points A, B, and C to different locations to illustrate the ideas of the chapter.
Questions you might ask in your role as student to your student:
What concepts are illustrated in the original drawing?
How could you move each of the points A, B, and C to show an inscribedangle?
How could you move each of the points A, B, and C to show tangent segments?
A
C
B
Radius
Diameter
Chord
Tangent
(continued)
DG4GP_905_06.qxd 12/27/06 10:23 AM Page 25
26 Discovering Geometry: A Guide for Parents 2008 Key Curriculum Press
Chapter 6 Discovering and Proving Circle Properties (continued)
How could you move points A, B, and C to show an angle inscribed in asemicircle?
How could you move points A, B, and C to show parallel lines interceptingcongruent arcs?
Sample Answers The original drawing shows a central angle, a sector, a chord, and an arc. If thecenter, C, is moved to lie on the circle, an inscribed angle is formed.
If C is moved to be outside the circle, and A and B are moved to make AC and BCtangents, then those segments are congruent. Or, the chord in the original diagramcould be rotated at one of its ends until it becomes a tangent segment.
If C is moved until AC is a diameter and B remains on the circle, ABC is a rightangle inscribed in a circle.
You would need to add another point and put C on the circle to show two chords.Parallel chords intercept equal arcs if they are equidistant from the center.
Of course many other answers are possible. Encourage your student to think ofmultiple ways that A, B, and C could be moved to illustrate these same concepts.
AC
B
DG4GP_905_06.qxd 12/27/06 10:23 AM Page 26
Chapter 6 Review Exercises
Name Period Date
1. (Lesson 6.1) Given tangent AB, find mOAB, mAOB,and mABO.
2. (Lessons 6.2, 6.3) Find the unknown measures or lengths.
3. (Lesson 6.3) ABC is an equilateral triangle. Find mAB.
4. (Lesson 6.4) Write a paragraph proof to prove the following:
Given: Circle A with diameters EC and BD.
Prove: ED BC
5. (Lessons 6.5, 6.7) Given that the circumference of circle A is 24 in., find the radius of the circle and the length of BDC.
6. (Lessons 6.1, 6.3) AB and BC are tangents to the circle as shown.AC ED. Find a and b.
C
B
D
E
A
15 m
a
b100
140
C
A
B120
D
B
C
D
AE
A
BC
a
c d
b10 cm
130
A B
C
O
310
2008 Key Curriculum Press Discovering Geometry: A Guide for Parents 27
DG4GP_905_06.qxd 12/27/06 10:23 AM Page 27
4. Diameters CE and BD on circle A intersect to formcongruent vertical angles, so mBAC mDAE.The measure of an arc equals the measure of itscentral angle. Therefore, mBC mED because themeasures of their central angles are equal.
5. C 2r 24 in.; therefore, r 12 in.
mBDC 360 120 240
length of BDC 2346
00 (24) 16 in.
6. mCD mAE a Parallel secants inter-cept congruent arcs.
100 a a 140 360 360 in a circle.
a 60 Solve.
b 15 m Tangent segments fromthe same point arecongruent.
1. mAOB 90 Tangent is perpendi-cular to radius.
mAC 360 310 50 360 in a circle.
mAOC mAC 50 Central angle isequal to arcmeasure.
50 90 mABO 180 Triangle sum.
mABO 40 Subtraction.
2. b 90 Diameter is perpen-dicular to the chord.
d 10 cm Diameter perpendi-cular to a chordbisects the chord.
c 180 130 50 180 in a semicircle.
a 12(130) Inscribed angles.
ma 65 Division.
3. AB BC AC Equilateral triangle.
mAB mBC mAC Congruent arcs ofcongruent chords.
mAB mBC mAC 360 360 in a circle.
mAB mAB mAB 360 Substitution.
3mAB 360 Combine like terms.
mAB 120 Division.
28 Discovering Geometry: A Guide for Parents 2008 Key Curriculum Press
S O L U T I O N S T O C H A P T E R 6 R E V I E W E X E R C I S E S
DG4GP_905_06.qxd 12/27/06 10:23 AM Page 28
SMART Notebook
Lesson 8-3
Tangents
Lesson 8-3: Tangents
29.psd
THEOREM #1:
Example:
Find the value of
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Lesson 8-3: Tangents
THEOREM #2:
If two segments from the same exterior point are tangent to a circle, then they are congruent.
Example:
Find the value of
If AB = 1.8 cm, then AF = 1.8 cm
AE = AF + FE
AE = 1.8 + 7.0 = 8.8 cm
If FE = 7.0 cm, then DE = 7.0 cm
CE = CD + DE
CE = 2.4 + 7.0 = 9.4 cm
Lesson 8-3: Tangents
()
BCABBCAB
@=
^
ACDB
CEandAE
AC
222
2
2
34
916
25
5
AC
AC
AC
ACunits
+=
+=
=
=
SMART Notebook
Lesson 8-5
Angle Formulas
Lesson 8-5: Angle Formulas
185.psd
Central Angle
Central Angle
(of a circle)
Central Angle
(of a circle)
NOT A Central Angle
(of a circle)
An angle whose vertex lies on the center of the circle.
Definition:
Lesson 8-5: Angle Formulas
Central Angle Theorem
The measure of a center angle is equal to the measure of the intercepted arc.
Intercepted Arc
Center Angle
Example:
Give is the diameter, find the value of x and y and z in the figure.
Lesson 8-5: Angle Formulas
Example: Find the measure of each arc.
4x + 3x + (3x +10) + 2x + (2x-14) = 360
14x 4 = 360
14x = 364
x = 26
4x = 4(26) = 104
3x = 3(26) = 78
3x +10 = 3(26) +10= 88
2x = 2(26) = 52
2x 14 = 2(26) 14 = 38
Lesson 8-5: Angle Formulas
Inscribed Angle
Inscribed Angle: An angle whose vertex lies on a circle and whose sides are chords of the circle (or one side tangent to the circle).
1
4
2
3
No!
No!
Yes!
Yes!
Examples:
Lesson 8-5: Angle Formulas
Intercepted Arc
Intercepted Arc: An angle intercepts an arc if and only if each of the following conditions holds:
1. The endpoints of the arc lie on the angle.
2. All points of the arc, except the endpoints, are in the interior of the angle.
3. Each side of the angle contains an endpoint of the arc.
Lesson 8-5: Angle Formulas
Inscribed Angle Theorem
The measure of an inscribed angle is equal to the measure of the intercepted arc.
Y
Z
55
110
Inscribed Angle
Intercepted Arc
An angle formed by a chord and a tangent can be considered an inscribed angle.
Lesson 8-5: Angle Formulas
Examples: Find the value of x and y in the fig.
Lesson 8-5: Angle Formulas
An angle inscribed in a semicircle is a right angle.
R
P
180
S
90
Lesson 8-5: Angle Formulas
Interior Angle Theorem
Angles that are formed by two intersecting chords.
Definition:
The measure of the angle formed by the two intersecting chords is equal to the sum of the measures of the intercepted arcs.
Interior Angle Theorem:
E
2
Lesson 8-5: Angle Formulas
A
B
C
D
x
91
85
Example: Interior Angle Theorem
y
Lesson 8-5: Angle Formulas
Exterior Angles
An angle formed by two secants, two tangents, or a secant and a tangent drawn from a point outside the circle.
Two secants
A secant and a tangent
2 tangents
Lesson 8-5: Angle Formulas
Exterior Angle Theorem
The measure of the angle formed is equal to the difference of the intercepted arcs.
Lesson 8-5: Angle Formulas
Example: Exterior Angle Theorem
Lesson 8-5: Angle Formulas
100
30
25
Lesson 8-5: Angle Formulas
Inscribed Quadrilaterals
mDAB + mDCB = 180
mADC + mABC = 180
If a quadrilateral is inscribed in a circle, then the opposite angles are supplementary.
Lesson 8-5: Angle Formulas
int.
AECandDEBareeriorangles
92
y
=
o
1
()
2
1
(9185)
2
xmACmDB
x
=+
=+
oo
.
InthegivenfigurefindthemACB
int.
ADCistheerceptedarcofABC
88
x
=
o
18088
y
=-
oo
1
()
2
1
(26595)
2
1
(170)85
2
mACBmADBmAD
mACB
mACB
=-
=-
==
oo
oo
265
95
C
B
A
O
B
A
C
D
.
ABCisaninscribedangle
AD
z
25
55
y
x
O
B
D
A
C
3x+10
2x-14
2x
4x
3x
B
D
C
E
A
25
180(2555)18080100
55
x
y
z
=
=-+=-=
=
o
oooo
o
2
mAB
mABC
=
A
C
B
D
50100
2
100
50
22
mAC
mA
y
C
mAC
x
=
==
===
o
o
40
20
22
40
50
22
1004060
mAD
mADmD
y
y
Cy
x
===
++
==
=+=
o
o
3
y
x
2
y
x
1
x
y
12
2
mACmDB
mm
+
==
1
2
xy
m
-
=
oo
3
2
xy
m
-
=
oo
2
2
xy
m
-
=
oo
A
B
C
D
180
2100
155
3()22.5
22
180155
4()117.5
22
5180117.562.5
11
6()(10030)35
22
mmFG
mmAG
mmCEmEF
mmGFmACE
m
mmAGmCE
==
==
=+==
+
=+==
=-=
=-=-=
o
o
o
o
ooo
o
,100,3025.
.
GivenAFisadiametermAGmCEandmEF
Findthemeasureofallnumberedangles
===
ooo
SMART Notebook
Lesson 8-3
Tangents
Lesson 8-3: Tangents
29.psd
THEOREM #1:
Example:
Find the value of
If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency.
Lesson 8-3: Tangents
THEOREM #2:
If two segments from the same exterior point are tangent to a circle, then they are congruent.
Example:
Find the value of
If AB = 1.8 cm, then AF = 1.8 cm
AE = AF + FE
AE = 1.8 + 7.0 = 8.8 cm
If FE = 7.0 cm, then DE = 7.0 cm
CE = CD + DE
CE = 2.4 + 7.0 = 9.4 cm
Lesson 8-3: Tangents
()
BCABBCAB
@=
^
ACDB
CEandAE
AC
222
2
2
34
916
25
5
AC
AC
AC
ACunits
+=
+=
=
=
SMART Notebook
Lesson 8-4
Arcs
and Chords
Lesson 8-4: Arcs and Chords
74.psd
Theorem #1:
In a circle, if two chords are congruent then their corresponding minor arcs are congruent.
Example:
Lesson 8-4: Arcs and Chords
Theorem #2:
In a circle, if a diameter (or radius) is perpendicular to a chord, then it bisects the chord and its arc.
Example:
If AB = 5 cm, find AE.
Lesson 8-4: Arcs and Chords
Theorem #3:
In a circle, two chords are congruent if and only if they are equidistant from the center.
Example:
If AB = 5 cm, find CD.
Since AB = CD, CD = 5 cm.
Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a chord that is 15 inches long and 8 inches from the center of the circle.
Draw a radius so that it forms a right triangle.
How could you find the length of the radius?
ODB is a right triangle and
Solution:
x
Lesson 8-4: Arcs and Chords
Try Some Sketches:
Draw a circle with a diameter that is 20 cm long.
Draw another chord (parallel to the diameter) that is 14cm long.
Find the distance from the smaller chord to the center of the circle.
14 cm
x
E
Solution:
OB (radius) = 10 cm
EOB is a right triangle.
7.1 cm
Lesson 8-4: Arcs and Chords
IfABCDthenABCD
@@
127,.
GivenmABfindthemCD
=
o
CDABiffOFOE
@@
sec
ODbitsAB
14
sec.7
22
AB
OEbitsABEBcm
\===
222
222
2
107
1004951
51
OBOEEB
X
X
X
=+
=+
=-=
==
sec.
^
\@@
IfDCABthenDCbitsABandAB
AEBEandACBC
127
SincemABmCD
mCD
=
=
o
5
2.5
22
120
,60
22
AB
AEAEcm
mAB
mACmAC
=\==
=\==
o
222
222
AB15
DB===7.5cm
22
OD=8cm
OB=OD+DB
OB=8+(7.5)=64+56.25=120.25
OB=120.2511
cm
120,.
IfmABfindmAC
=
o
SMART Notebook
Lesson 8-6
Segment Formulas
Lesson 8-6: Segment Formulas
39.psd
Intersecting Chords Theorem
Interior segments are formed by two intersecting chords.
If two chords intersect within a circle, then the product of the lengths of the parts of one chord is equal to the product of the lengths of the parts of the second chord.
a
b
c
d
a b = c d
Theorem:
Lesson 8-6: Segment Formulas
Intersecting Secants/Tangents
Exterior segments are formed by two secants, or a secant and a tangent.
Two Secants
Secant and a Tangent
Lesson 8-6: Segment Formulas
Intersecting Secants Theorem
a e = c f
If two secant segments are drawn to a circle from an external point, then the products of the lengths of the secant and their exterior parts are equal.
Lesson 8-6: Segment Formulas
Example:
x
6 cm
2 cm
4 cm
AB AC = AD AE
4 10 = 2 (2+x)
40 = 4 + 2x
36 = 2x
X = 18 cm
Lesson 8-6: Segment Formulas
Secant and Tangent Theorem:
a
b
c
a2 = b d
d
The square of the length of the tangent equals the product of the length of the secant and its exterior segment.
Lesson 8-6: Segment Formulas
Example:
x
9 cm
25 cm
Lesson 8-6: Segment Formulas
9,25..
InthefigureifADcmandACcmFindx
==
2
2
925
22515
ABADAC
x
xcm
=
=
==
;6,2,4..
InthefigureifBCcmADcmABcmFindx
===
C
B
D
f
e
d
c
b
a
A
E
SMART Notebook
Lesson 9-4
Spheres
Lesson 9-4: Spheres
10.psd
Spheres
A sphere is formed by revolving a circle about its diameter.
In space, the set of all points that are a given distance from a given point, called the center.
Definition:
Lesson 9-4: Spheres
Spheres special segments & lines
Radius: A segment whose endpoints are the center of the sphere and a point on the sphere.
Chord: A segment whose endpoints are on the sphere.
Diameter: A chord that contains the spheres center.
Tangent: A line that intersects the sphere in exactly one point.
Radius
Chord
Diameter
Tangent
Lesson 9-4: Spheres
Surface Area & Volume of Sphere
Volume (V) =
Surface Area (SA) = 4 r2
Example:
Find the surface area and volume of the sphere.
12 cm
Lesson 9-4: Spheres
Great Circle & Hemisphere
Great Circle: For a given sphere, the intersection of the sphere and a plane that contains the center of the sphere.
Hemisphere: One of the two parts into which a great circle separates a given sphere.
Great Circle
Hemisphere
Lesson 9-4: Spheres
Surface Area & Volume of Hemisphere
Find the surface area and volume of the following solid (Hemisphere).
10 cm
Lesson 9-4: Spheres
3
4
3
r
p
22
33
.412576
4
122304
3
SAcm
Vcm
pp
pp
==
==
gg
gg
2
2
3
3
.410400
400
.200
2
4
101333.3
3
1333.3
666.7
2
SAofSphere
SAofHemispherecm
VofSphere
VofHemispherecm
pp
p
p
pp
p
p
==
==
==
==
gg
gg
SMART Notebook
Chapter 8: Circles Name ________________________ Section 8-5: Angle Formulas Date ___________ Homework Period _________
#5#4
#3#2#1
162x
A
B
C
C B
A
x
98
x10945
x
60 x
CB
A
C
B
A A
B
C
#10 #9
#8#7 #6
C
B
A
85
x
C
B
A
C
BA x
272
140x
A
BC
C B
A
x
180
20m ABC =
Find AB
E
EE
E E
150
x
85
135 60
140
30
97
C
B
A
D x 50
#14 #15
B 115
D
A
C
B
x D
A
C C
A D
x
B
#13#12#11
30
x D
A
B
C
x 91
#16
EE
E
E E
125
60 90170
115
70
185
CB
A
#19 #20
B
D
A
C
Bx
D
A
C
C
A
Dx
B
#18#17
25 x
D
A
B
C
P
20m BEC =
Find BC
SMART Notebook
E
D
B
A C E
D
B
A CE
D
B
A C
BC
E
F
D
A
E
B
G
A
C
D F E
B
G
A
C
D F
E
D
B
A C
E
B
G
A
C
D F
Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Classwork Period _________ 1. 2. 3.
AC BD AC BD AC BD m 94ABC = m 4AE = m 12AC =
Find AB ______ Find AC ______ m 8DE = Find the radius______ 4. 5. 6.
Find AB ______ GB GE GB GE Find ABF ______ m 10EF = m 5EF = Find ABD ______ Find DF _______ Find mCA = ____
7. 8.
AC BD 17DA = AC DF
m 8ED = 100mAC = Find AC ______ Find mDF _____
9. Suppose a chord is 9 meters from the center of a circle. It is 20 meters long. Find the length of the radius. ___________
10. Find the length of a chord 4 inches from the center of a circle with a radius of 5 inches. __________
SMART Notebook
E
D
B
A CE
D
B
A C E
D
B
A C
E
B
G
A
C
D F E
B
G
A
C
D FE
B
G
A
C
D F
C
D E
B
Chapter 8: Circles Name ________________________ Section 8-4: Arcs and Chords Date ___________ Homework Period _________ 1. 2. 3.
AE EC AC BD AC BD Find m AEB ______ m 10AC = m 22ED = Find m AE ______ DC = 32 Find m EB _____ 4. 5. 6. AC DF AC DF mGE = 7 80mAF = m BG = 4 GF = 25 60mCD = Find mGE _______ Find m DF ______ Find mAC _______ 7. 8. Suppose that a circle has a radius of 35 units and a chord
is 56 units. Find the distance from the center to the chord. __________.
210mBE = Find mCD _____ 9. Suppose the diameter of a circle is 20 feet long and a
non-intersecting chord is 12 feet long. Find the distance between the chord and the center. __________
SMART Notebook
Chapter 8: Circles name ________________________ Lesson 8-1: Terminology date ______________ Classwork period _____ Sketch Define
1. circle ____________________________________________________ ___________________________________________________________ 2. radius ___________________________________________________ ___________________________________________________________ 3. chord ____________________________________________________ ___________________________________________________________ 4. diameter __________________________________________________ ___________________________________________________________ 5. secant ____________________________________________________ ___________________________________________________________ 6. tangent ___________________________________________________ ___________________________________________________________ 7. point of tangency ___________________________________________ ___________________________________________________________ 8. common tangent ___________________________________________ ___________________________________________________________ 9. congruent circles ___________________________________________ ___________________________________________________________ 10. concentric circles __________________________________________ ___________________________________________________________
11. inscribed ________________________________________________ ___________________________________________________________ 12. circumscribed ____________________________________________ ___________________________________________________________ 13. arc _____________________________________________________ ___________________________________________________________ 14. minor arc ________________________________________________ ___________________________________________________________ 15. semicircle ________________________________________________ ___________________________________________________________ 16. major arc ________________________________________________ ___________________________________________________________ 17. arc length ________________________________________________ ___________________________________________________________ 18. circumference ____________________________________________ ___________________________________________________________ 19. sphere __________________________________________________ ___________________________________________________________ 20. great circle _______________________________________________ ___________________________________________________________
SMART Notebook
F
O
BA
E
C
G
D
O
Y
W
X
Z
Chapter 8: Circles Name____________________ Lesson 8-1: Terminology Date ______________ Homework Period ___ For questions 1 - 7 refer to the circle to the right. 1. Name the circle.______________ 2. Name all radii._______________ 3. Name a diameter.______________ 4. Name a chord._______________ 5. Name a tangent._______________ 6. Name a secant.______________ 7. Name a point of tangency.________________ For questions 8 - 13 refer to circle to the right. 8. XY is a ____________ of circle O. 9. XO is a ____________ of circle O. 10. XY is a _____________ of circle O. 11. WZ appears to be ____________ to circle O. 12. XYZ is ___________ in circle O. (Hint: X, Y, and Z lie on circle O) 13. XYZ is a ________ _________ of circle O. For questions 14-16, complete. 14. Congruent circles have ____________ radii. 15. A secant of a circle is a(n) ___________ that intersects a circle at exactly________ point(s). 16. Concentric circles have the same _____________. Determine whether each statement is true or false. 17. A chord of a circle that passes through the center of the circle is called a diameter. 18. If two circles are concentric, then their diameters have equal measure.
SMART Notebook
E
B
A
C
D
C
O
A
B
D
B
D
P
E
C
A
Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Classwork Period ___ Secants, chords and tangents are shown. For questions 1 - 6, refer to the figure below and find the indicated value. 1. If CE = 3, DE = 6, and AE = 2, find BE. 2. If AE = 3, BE = 5, and DE = 2, find CE. 3. If AE = 3, BE =6 , and CE = 4, find DE. 4. AE = 12, BE = 18, and DE = 9, find CE. 5. If AE = 3.4, BE = 5.2, and CE = 2, find DE. 6. If AE = 2x, BE = 4x, CE = 8, and DE = 16, find x. For questions 7 - 11, refer to the figure below and find the indicated value. 7. If BC = 3 and BD = 12, find AB. 8. If AB = 6 and BD = 12, find BC. 9. If BC = 4 and CD = 12, find AB. 10. If AB = 6 and BD = 9, find BC. 11. If AB = 10 and BC = 5, find CD. For questions 12 - 21, refer to the figure below and find the indicated value. 12. If AC = 12, BC = 4 and CE = 8, find CD. 13. If CE = 9, CD = 4, and BC = 3, find AB. 14. If DE = 3, DC = 9 and BC = 6, find AB. 15. If AB = 17, BC = 3, And CD = 6, find CE. 16. If DE = 8, CD =7, and AC = 21, find BC. 17. If CE =15, DE = 10, BC = 4, find AB. 18. If CD = 8, DE = 10, and AB = 10, find BC. 19. If BC = 5, AB=7, CD = x and DE = 5x, find x. 20. If BC = 12, AB = 13, CD = x, and DE = 2x, find x.
SMART Notebook
B
O
E
D
A C
B
O
D
A C
B
OD
A
E
C
Chapter 8: Circles Name____________________ Lesson 8-6: Segment Formulas Date ______________ Homework Period ___ For questions 1 - 6, refer to the figure below and find the indicated value. 1. If AB = 25, BC = 3, and BE = 15, find BD. 2. If AB = 4, BC = 9, and BD = 6, find BE. 3. If AC = 16, AB = 4, and BE = 8, find DE. 4. If DE = 17, BD = 7, and AB = 5, find AC. 5. If AB = 3, BC = 5 and BE = 8, find BD. 6. If BE = 16, BD = 4, and B is the midpoint of AC, find AB. In the accompanying diagram, * is tangent to circle O at D and * is a secant. 7. If AD = 9 and AB = 3, find AC. 8. If BC = 15 and AB = 1, find AD. 9. If AD = 8 and AB = 4, find AC. 10. If AB = 4 and BC = 5, find AD. 11. If AD = 3 5 and AB = 3, find BC. In the accompanying diagram, two secants are drawn from the same point. 12. If AB = 5, AC = 8, and AD = 2, find DE. 13. If AB = 3, BC = 7 and AE = 15, find AD. 14. If AB = 6, BC = 12, and AD = 4, find DE. 15. If AC = 20, AD = 8, and DE = 2, find AB. 16. If AB = 5, AD = 8 and DE = 2, find BC. 17. If B is the midpoint of AC , and AD = 8, and DE = 17, find AC.
SMART Notebook
Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Classwork period ___
Fill in the chart using the given information about spheres. Write your answers in terms of .
(256/3)
72
288
100
2
6
1/8
VolumeSurface AreaDiameterRadius
SMART Notebook
Chapter 9: Area and Volume name _____________________________Lesson 9-4: Spheres date ______________Homework period ___
Find the surface area of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .
1) radius is 12 cm
2) diameter is 16 m
3) diameter is 1 ft
4) radius is 9 in
5) radius is 6 mm
Find the volume of a sphere with the given radius or diameter. Express your answers in both decimal form and in terms of .
1) radius is 1 m
2) diameter is 18 ft
3) diameter is 16 cm
4) radius is 12 in
5) radius is 8 mm
SMART Notebook
x
12
BO
A
C
12
7CB
A
x 9
x
12
1616
x
918
Chapter 8: Circles Name _____________________________ Lesson 8-3: Tangents Date ______________ Classwork Period ___ Find x. Assume that segments that appear to be tangent are tangent lines. Round answers to the nearest tenth. 1. x = ____________ 2. x = ____________ 3. x = ____________ 4. x = ____________
5. x = ____________
10
4
3
O
F
AB
C
D E
24
O
E
A
B
D
C6. Assume points A, E, and D are tangent to circle O. Find BC . 7. Assume D, E and F are tangent to circle O. Find AC . If BD bisects AC , BD AC , AB =13, and AC = 24. Find the indicated values.
8. BE = _______ 9. DE = _______ 10. If AB = 12 and m A =30, find BE = _________ and AE = __________. 11. If BE = 5 and m B = 60, find AB = ________ and AE = ___________.
E
B
D
A C
SMART Notebook
D
HEB
A
C
B
O
CA
Chapter 8: Circles Name____________________ Lesson 8-3: Tangents Date ______________ Homework Period ___ For questions 1 - 3, O and R are centers of circles. Find the indicated value. 1. 2. 3.
OR = ________ AB = _______ AD = _____ BC = _____ Refer to the accompanying figure for questions 4 - 6. Find the indicated values. 4. If HD = 12 and EH = 9, DE = _____________. 5. If DE = 17 and BH = 9, CD = _____________. Refer to the accompanying figure for questions 7 - 10. * is tangent to circle O. 6. If AC = 4 and OC = 3, then AO = ___________. 7. If OC = 15 and AC = 20, then AO = _______________. 8. If m OAC =30 and AO = 10, then OC = ___________. 9. If m OAC = 60 and OC = 4 3 , then AC = _________.
62
O R 5OB1
RA1
43
109 BA
CD
O
DC
A
B
CD and BC are tangent to circle O. Refer to the accompanying figure and find the indicated values. 10. If OC = 20 and OD = 12, then BC = ___________. 11. If OC = 4 2 and CD = 4, then OD = ____________. 12. If AD = 10 and CD = 12, then OC = _____________. 13. If OC = 5 3 and CD = 5 2 , then AD = ____________. 14. If m OCD = 30 and OD = 6, then OC = ____________ and CD = ___________. 15. If m COD = 60 and CD = 4 3 , then OC = ___________ and AD = ____________.
SMART Notebook
Geometry/ Honors Geometry Instructional Guide Worksheets Unit 8 - Circles Page 16: Practice
MCPS 2005
Find the area of each shaded sector. 1. 2. 3. 4. 5. 6. 7. A sector has an area of 36 cm2. The radius of the sector is 6 cm. What is the degree measure of the intercepted arc? 8. A sector has an area of 12 in2. The measure of the intercepted arc is 120o. What is the radius of the circle?
3 cm 100o
5 cm
75o
16 in
135o 12 in
80o 9 mm 210o
18mm
60o
Geometry/ Honors Geometry Instructional Guide Worksheets Unit 8 - Circles Page 17: Practice
MCPS 2005
Practice: Find the area of the following shaded segments of circles. 1. 2. 3. 4. 5. 6.
7. Use right triangle trigonometry to help you find the area of this segment.
90o
20 cm
120o
9 cm
8 in
60o
60o
9 cm 120o
3 in
80o
5 cm
90o
6 cm
Geometry/ Honors Geometry Instructional Guide Worksheets Unit 8 - Circles Page 18: Examples
MCPS 2005
A locus (pl. loci) is a set of points, all of which meet a stated condition. To sketch a locus, draw points of the locus until you see a pattern.
Examples: Draw and describe the following loci:
1. In a plane, the points 2 cm from a given point P.
2. In a plane, the points equidistant from two parallel lines.
3. In a plane, the points equidistant from two given points.
4. In a plane, the points 3 cm from a given line segment.
5. Describe the loci in the first four problems, when the word plane is replaced with space.
Geometry/ Honors Geometry Instructional Guide Worksheets Unit 8 - Circles Page 18: Practice
MCPS 2005
Draw a sketch and write a description for each of the following.
1. The locus of points in a plane that are 4 units from a given line. 2. The locus of points in a plane that are equidistant from two parallel lines which
are 5 cm apart. 3. The locus of points in a plane 10 cm from point A. 4. The locus of points in your bedroom equidistant from the ceiling and the floor. 5. The locus of points on a football field equidistant from both goal lines. 6. The locus of points equidistant from the vertices of triangle ABC. 7. The locus of points that are equidistant from the sides of triangle ABC.
Write a description for each of the following.
8. The locus of points in space 5 cm from a given point. 9. The locus of points in space equidistant from given two points.
10. The locus of points in space 5 cm from a given line.
11 The locus of points in space equidistant from the vertices of a square.
SMART Notebook
Name ___________________________________________________ Date ______________
Math Worksheet Center
Quiz: Area of Sector and Segment
1
Find the area of shaded sector shown in fig. The radius of the circle is 110 units and the length of the arc measures 30 units.
2
Find the area of a segment of a circle if the central angle of the segment is 170 and the radius is 34.
3
There are 3 equal segments in semi-circle O. If the radius of circle is 15, what is the area of each segment?
4
Find the area of the shaded sector of circle O. The radius is 90inches and the central angle is 160.
5
Find the area of a segment of a circle if the central angle of the segment is 80 and the radius is 320.
6
Find the area of shaded sector shown in fig. The radius of the circle is 225 units and the length of the arc measures 170 units.
7
Find the area of the shaded sector of circle O. The radius is 50inches and the central angle is 40.
8
There are 3 equal segment sin semi-circle O. If the radius of circle is 102, what is the area of each segment?
9
Find the area of shaded sector shown in fig. The radius of the circle is 35 units and the length of the arc measures 35 units.
10
There are 3 equal segments in semi-circle O. If the radius of circle is 20, what is the area of each segment?
Circle # Correct 0 1 2 3 4 5 6 7 8 9 10
Percentage Score 0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
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MA1G5b.
By: Allen Moore
STANDARD
Determine the effect on surface area and volume of changing the radius or diameter of a sphere.
Use
This standard can be used to show how changing the radius or diameter of a basketball will effect its surface area and volume. This is important because changing the radius or diameter of a basketball will determine whether or not the ball will fit through the basketball goal.
RELEVANT INFORMATION:
SURFACE AREA OF A SPHERE
SURFACE AREA
CHANGING RADIUS:
y=4x
x= radius
y= surface area
CHANGING DIAMETER:
y=4(x/2)
x= diameter
y= surface area
Surface Area of Spheres
*Double radius to find surface area by diameter
RadiusDiameterSurface Area
1212.57
2450.27
36113.1
48201.06
510314.16
612452.39
714615.75
816804.25
9181017.88
10201256.64
RELEVANT INFORMATION:
VOLUME OF A SPHERE
VOLUME
CHANGING RADIUS:
y=4/3x
x= radius
y= volume
CHANGING DIAMETER:
y=4/3(x/2)
x= diameter
y= volume
Volume of Spheres
*Double radius to find volume by diameter
Radius
Volume
RadiusDiameterVolume
124.19
2433.51
36113.1
48268.08
510523.6
612904.78
7141436.76
8162144.66
9183053.63
10204188.79
Regulation NBA Dimensions
Regulation NBA size basketball:
Radius: 4.695 in.
Surface Area: 277 in.
Volume: 433.51 in.
Regulation hoop size:
Radius: 9 in.
Area: 254.47 in.
How much can the size of a basketball increase and still fit through the regulation hoop?
The basketballs radius can increase in size by about 3.034 in. and still fit through the hoop.
The size of the new basketball would be:
Radius: 8.999
Volume: 3052.61 in.
Surface Area: 1017.65 in.
How much can the size of a basketball increase and still fit through the regulation hoop?
(continued)
Area of regulation hoop:
254.47 in.
Area of circle of new sized basketball:
254.41 in.
This relates to real life, because if the size of a regulation NBA basketball is increased, it would be harder to make a shot and score points in a basketball game. If the ball is larger, there is less probability of making a shot in a game and scoring points. This would decrease the total score in games. Therefore, records and other outcomes of basketball games would be different.
Effects of a different sized basketball
For example, if players practiced with a basketball with a radius of 4 in., then they would not be prepared for a game because they would not be accustomed to a regulation size ball (radius= 4.695 in.), which they would be required to use during a game. They would have practiced with a basketball with 165.43 in. less than what is used for competitions, and nearly (201.06 in.) less than the regulation surface area of a basketball (277 in.). If these players compete in games with a regulation sized ball, they would not perform to their potential because of their lack of experience with a regulation sized ball.
Final Effects
Whether the radius of a basketball is increased or decreased, drastic effects could still occur. If the radius was decreased, there would be less surface area to make rebounds and catch passes, and less volume which would effect the distance of shots and passes. Likewise, if the radius were increased, it would take more strength to make a normal shot and score, and to some extent, the ball might not fit through the hoop.
3D Example of a Basketball
Citations
"Circle and Sphere Calculator." 2008. CSG, Computer Support Group, Inc. and CSGNetwork.Com . 14 Dec 2008 .
"Create a Graph." Welcome to the NCES Kids' Zone. 14 Dec 2008 .
"What is the diameter circumference or radius of a basketball?." Wiki Answers. 2008. Answers Corporation. 14 Dec 2008 .
4.19
33.51
113.1
268.08
523.6
904.78
1,436.76
2,144.66
3,053.63
4,188.79
0
900
1,800
2,700
3,600
4,500
12345678910
12.57
50.27
113.1
201.06
314.16
452.39
615.75
804.25
1,017.88
1,256.64
0
260
520
780
1,040
1,300
12345678910
Radius
Surface
Area
SMART Notebook
MA1G4. Students will understand the properties of circles.a. Understand and use properties of chords, tangents, and secants as an application of triangle similarity.b. Understand and use properties of central, inscribed, and related angles.c. Use the properties of circles to solve problems involving the length of an arc and the area of a sector.d. Justify measurements ad relationships in circles using geometric and algebraic properties.
MA1G5. Students will find and compare the measures of spheres.a. Use and apply surface area and volume of a sphere.b. Determine the effect on surface area and volume of changing the radius or diameter of a sphere.
Accelerated Math IUnit3
Unit 3 Circles and Spheres
New Learning for Students Essential QuestionsWhat are the properties of chords, tangents,and secants of a circle?How do you use the properties of chords,tangents, and secants as an application oftriangle similarity?How can you use properties of central,inscribed, and related angles?How can you use properties of circles to solveproblems involving the length of an arc andthe area of a sector?How can you justify measurements andrelationships in circles using geometric andalgebraic properties?How do you calculate surface area andvolume of a sphere?How is the surface area and volume of asphere altered when the radius is changed?
StandardsKEYMA1G4. Students will understand the propertiesof circles.
a. Understand and use properties of chords,tarigents, and secants as an application oftriangle similarity.
b. Understand and use properties of central,inscribed, and related angles.
c. Use the properties of circles to solve problemsinvolving the length of an arc and the area of asector.
d. Justify measurements and relationships incircles using geometric and algebraicproperties.
MA1G5. Students will find and compare themeasures of spheres.
a. Use and apply surface area and volume of asphere.
b. Determine the effect on surface area andvolume of changing the radius or diameter of asphere.
Related All process standards
Textbook CorrelationsMcDougal Littell (Math 2): 6.1-6.9
VocabularyChordTangentSecantCentral angleInscribed angleRelated angleArc lengthSectorSphere
SMART Notebook
1. Determine the area of a sector of a circle of radius 9 cm intercepted by a central angle of 120. (Nearest tenth)
2. Determine the area of a sector of a circle of radius 10 in intercepted by a central angle of . (Nearest tenth)
90
o
1
4
90
o
901
3604
=
o
o
m
o
80
o
1201
3603
=
o
o
1
3
120
o
2
19.6
in
=
rad
n
802
3609
=
o
o
?
2
9
2.5
2
p
360
m
o
o
2.5rad
80
o
2.5
2
p
2.5rad
2
n
p
360
m
o
o
m
o
2
9
p
r
2
n
p
120
o
2
r
p
2
r
p
8
p
8
2
p
p
2
84.8cm
=
2
10
p
120
360
o
o
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001213 Name ____________________________PMa. 11 Block _____
7.6 Arc Length and Sector Area Worksheet
Find the length of the indicated arc, to the nearest tenth.
1.
15 cm71
2.
12 m
149
Find the area of the indicated sector, to the nearest tenth.
3.
17 m
221
4.
35.2 cm
40
001212 7.5 Tangents to a Circle Worksheet
- 2 -
Find the radius. Round the nearest tenth.
5.
19.6 cm
130
6.12 cm
17
7.
A= 165.6 cm 2
129
8.
A= 31.2 cm 2256
001212 7.5 Tangents to a Circle Worksheet
- 3 -
9.
3.9 cm
7.1 cm
10.
56.9 m15 m
11.
9.5 mA = 98.6 m 2
12.
A = 1720 cm 2
65 cm
001212 7.5 Tangents to a Circle Worksheet
- 4 -
13. The cross section of seven circular metal rods bound together by a strap is shown. If the diameter of each rod is 1.8 cm, and 2.5 cm of strap is required to fasten the ends of the strap, how long is the strap? Round to the rearest tenth.
14. A rectangular pizza, 40 cm by 60 cm, is cut into 24 square pieces. Two round pizzas, each cut into 12 slices, also give 24 pieces. So that the pizzas are the same size, what must be the diameter of the round pizzas?
001212 7.5 Tangents to a Circle Worksheet
- 5 -
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