geometry unit 10: circles geometry unit 10: circles - weebly
TRANSCRIPT
Geometry Unit10:Circles
Ms.Talhami 1
GeometryUnit10:Circles
Name_________________
Geometry Unit10:Circles
Ms.Talhami 2
HelpfulVocabularyWord Definition/Explanation Examples/HelpfulTips
Geometry Unit10:Circles
Ms.Talhami 3
EquationofaCircle
Determinethecenterandradiusofthegivencircles:1.(x ā 7)2 + (y + 10)2 = 81 center______ radius___
2. (x + 5)2 + (y + 1)2 =25 center______ radius___
3. (x ā 4)2 + (y + 5)2 = 9 center______ radius___
4. (x + 9)2 + (y ā 5)2 = 12 center______ radius___
5. (x ā 1)2 + (y ā 5)2 = 16 center______ radius___
6. (x + 5)2 + (y + 6)2 = 9 center______ radius___
7.
A) center_____radius______ equation_________________ B) center_____radius______ equation_________________
8.
A) center_____radius______ equation_________________ B) center_____radius______ equation_________________
B
A
B
A
Geometry Unit10:Circles
Ms.Talhami 4
Graphthefollowingcircles.Statethecenterandradius.1. a) (x ā 3)2 + (y + 2)2 = 4 b) (x + 6)2 + (y - 7)2 = 9
2. a) x2 + (y - 7)2 = 25 b) (x - 4)2 + (y + 5)2 = 9
StandardFormandPerfectSquareTrinomials1.(xā2)2a=______b=______c=______
2.(x+5)2a=______b=______c=______
3.(xā9)2a=______b=______c=______
CompletingtheSquareDeterminethevalueoftheconstantterm,c,tocreateaperfectsquaretrinomialthenwritethetrinomialinfactoredform.1.
x2+4x+___FactoredForm_____________
2.x2+10x+___
FactoredForm_____________
3.x2+14x+___
FactoredForm_____________
4.x2ā12x+___
FactoredForm_____________
5.x2ā8x+___
FactoredForm_____________
6.x2ā2x+___
FactoredForm_____________
Geometry Unit10:Circles
Ms.Talhami 5
DoNowFinding the Equation of a Circle 1. Circle A center_____ radius______ equation _________________
2. Circle B center_____ radius______ equation _________________ 3. Circle O center_____ radius______ equation ________________
UsingCompletingtheSquarewithQuadraticEquationstoRewritefromStandardFormtoVertexForm1.
x2+6x+3=0
2.x2+10x+20=0
3.x2ā8xā3=0
UsingCompletingtheSquarewithCircleEquationstoRewritefromStandardFormtoCenterRadiusFormDeterminethecenterandradiusofthegivencircles1.
x2+y2+4xā16y+52=0
2.x2+10x+y2ā16=0
Geometry Unit10:Circles
Ms.Talhami 6
PracticeDeterminethecenterandradiusofthegivencircles1.
x2+y2+2x+18y+1=0
2.x2+y2+18x+17=0
3.x2ā14x+y2ā2yā50=0
4.x2+y2ā10x+10y=ā48
CircleVocabulary&Activity
Geometry Unit10:Circles
Ms.Talhami 7
DoNowUsingcorrectcirclevocabulary,nameeachline:AC=HK=GD=GH=FB=ED= ArcMeasure
CentralAngle
C
A
B
C
A
BD
A
E 95Ā°
65Ā°
F
GH
x x x x
B
A C
B
A C
Geometry Unit10:Circles
Ms.Talhami 8
InscribedAngle
Practice 1. In a circle O, mā AOB = 87, mā BOC = 93, and mā COD = 35. Find the measure of each of the following: a) ā DOA_____ b) _____ c) _____ d) _____ e) _____ f) _____ g) _____ h) _____ i) _____
2. Given Circle B with diametersHC , EGand DA .
a) mā DBH = ___________ b) = ___________ c) = ___________ d) = ___________ e) mā HBA = ___________ f) mā DBA = ___________
InterceptedArc
A
B
C
F
E
GA
DC
B
22Ā°61Ā° 38Ā° A
GH
BFD
E C
Geometry Unit10:Circles
Ms.Talhami 9
3. Triangle ABC is inscribed in a circle, mA = 80, m = 88. Find: a) m _______ b) mā B __________ c) mā C________ d) m ___________ 4. Diameter ā„ chords at F, is a diameter and is a chord of circle O. If m = 60, find: a) m ______b) mā A_____c) mā C ________ d) m ______e) mā AOD____f) m ______ ExtraPractice1.
mā 1 = _______ mā 2 = _______
2.
mā 1 = _______ mā 2 = _______
3.
mā 1 = _______ mā 2 = _______
4.
mā 1 = _______ mā 2 = _______
mā 3 = _______
50Ā°
2
1
37Ā°
21 120Ā°
2
150Ā°
20Ā°32
1
Geometry Unit10:Circles
Ms.Talhami 10
5.
mā 1 = _______ mā 2 = _______ mā 3 = _______
6.
mā 1 = _______ mā 2 = _______
7.
mā 1 = _______ mā 2 = _______
8.
mā 1 = _______ mā 2 = _______ mā 3 = _______
9.
mā 1 = _______ mā 2 = _______
10.
mā 1 = _______ 2m = _______
11.
mā 1 = _______ mā 2 = _______
12.
mā 1 = _______ mā 2 = _______ mā 3 = _______
13.
1m = _______
14.
mā 1 = _______ 2m = _______
15.
mā 1 = _______ mā 2 = _______ mā 3 = _______
16.
mā 1 = _______ mā 2 = _______ 3m = _______
35Ā° 3
2
1
E 64Ā°
21
33Ā°
21
74Ā°
3
21
30Ā°2
1116Ā°
2
1
2
1
50Ā° 80Ā°
32
1
1 60Ā°
82Ā°
2
1 2
60Ā°
3 100Ā°
12
3
124Ā°
1
Geometry Unit10:Circles
Ms.Talhami 11
TakeitaStepFurther1.
x = __________ mā ADC = ___________
2.
x = __________ mā ABC = ___________
3.
x = __________ mā ACB = ___________
4. Given concentric circles with mGF = 76Ā°, mā HIE = 147Ā° and CA& FH are diameters.
= __________ = __________
= _________ mā CIB = _________
5. Given concentric circles with mBC = 31Ā°, mā FKJ = 68Ā° and EB is a diameter.
= __________ mā GKH= __________
= _________ mā AKB = _________
x+34
4x+8
B C
D
A
5x-81
2xB
FC
D
A
x-25
x+34C
B
A
E
G
H
C
I
D
AB
F
31Ā°
68Ā°
H
J I
F
G
B
K
E D
A
C
Geometry Unit10:Circles
Ms.Talhami 12
AnglesFormedInsideandOutsideaCircleAnglesFormedONaCircle
AnglesFormedINSIDEaCircle
1 2
G
B
DR
112Ā° 1
x
78Ā°
1
y
118Ā°
x
CA
F
E
B
D
x
107Ā°
55Ā°
x71Ā°92Ā°
82Ā°
x
146Ā°
Geometry Unit10:Circles
Ms.Talhami 13
AnglesFormedOUTSIDEaCircle
Practice 1.
mā 1 = _________
2.
mā 1 = _________
3.
mā 1 = _________
4.
mā 1 = _________
5.
1m = _________
6.
1m = _________
7.
mā 1 = _________
8.
x = ________
29Ā°x 93Ā°59Ā° x
168Ā° y
93Ā°
x
183Ā°
195Ā°92Ā° 1
95Ā°92Ā° 1
53Ā°
132Ā°
37Ā°
1
116Ā°
143Ā°
1
116Ā°
128Ā°1
133Ā°81Ā°
1
128Ā°
9x-10
5x+8
132Ā°
Geometry Unit10:Circles
Ms.Talhami 14
9.
mā 1 = _____ mā 2 = _____
10.
mā 1 = _____ 2m = _____
11.
1m = _____
mā 2 = _____
12.
1m = _____
mā 2 = _____
13.
mā 1 = _________
14.
mā 1 = _________
15.
mā 1 = _________
16.
mā 1 = _________
17.
1m = ________
18.
1m = ________
19.
1m = ________
20.
1m = ________
ExtraPractice1.
mā 1 = _____ mā 2 = _____
2.
mā 1 = _____ 2m = _____
3.
mā 1 = _____ mā 2 = _____
21102Ā°
2
1 79Ā° 2
1
50Ā°
72Ā°2
1
1 48Ā°
132Ā°
151Ā°
168Ā°1 78Ā°
158Ā°
1128Ā°
160Ā°
1 39Ā°
134Ā°
1
39Ā°28Ā°
143Ā°
149Ā°
62Ā°1
28Ā°
263Ā° 75Ā°
1
266Ā°
2
1
263Ā° 75Ā°
1
Geometry Unit10:Circles
Ms.Talhami 15
4.
mā 1 = ______
5.
mā 1 = ______ mā 2 = ______
6.
mā 1 = ______ mā 2 = ______
7.
1m = ______
8.
mā 1 = ______ mā 2 = ______
9.
mā 1 = ______ 2m = ______
10.
x = ______ mā ABC = _______
11.
x = __________
12.
x = __________
13.
1m = ______ mā 2 = ______
14.
mā 1 = ______ mā 2 = ______
15.
mā 1 = ______
21Ā°1
168Ā°
128Ā°
66Ā°
21
50Ā°
2
94Ā°
1
55Ā°32Ā°1 2
1
58Ā°
128Ā°
66Ā°
21
7x-145x+8
E
C
A
B D
2x+34
59Ā°
73Ā° 4x+656Ā°
2
113Ā°
134Ā°
1 63Ā°42Ā°2
11
78Ā°
Geometry Unit10:Circles
Ms.Talhami 16
16.
mā 1 = ______ 2m = ______
17.
1m = ______ mā 2 = ______
18.
mā 1 = ______ mā 2 = ______
19.
mā 1 = ______ mā 2 = ______
20.
1m = ______ mā 2 = ______
21.
mā 1 = ______ mā 2 = ______
22. Solve for the missing values. a) mā 1 = _________ b) mā 2 = _________ c) mā 3 = _________ d) 4m = _________ e) mā 5 = _________ f) mā 6 = _________
40Ā°
2
44Ā°
1 2 58Ā°
1 28Ā°
2 1
2
35Ā°
1 160Ā°
73Ā°
2
75Ā°
134Ā°1
18Ā°
90Ā°
28Ā°
21
100Ā°
46 5
3
2
130Ā°
Geometry Unit10:Circles
Ms.Talhami 17
CongruentChords/Arcs
ParallelChords
ChordsandDiameters/Radii
A
E
C
D
B
A
E
C
D
B
F
E
B
D
AC
E
BAC
F
E
B
G
C
D
A
F
E
B
G
D
C
A
Geometry Unit10:Circles
Ms.Talhami 18
Practice1.
x = ___________
2.
AK = ___________
3.
x = ___________
4.
x = ___________
5.
x = ___________
6.
x = ___________
7.
x = ___________
8.
x = ___________
9.
x = ___________
10.
x = ___________
11.
x = ___________
12.
x = ___________
x
220Ā°
70Ā°
12cm
60Ā°
25Ā°
55Ā°
140Ā°B
A
H
J
I
K
78Ā°
91Ā°
113Ā°
9cm
10cm
11cm
x
x
27Ā°x
88Ā°
x
74Ā°
3.5cm
x
5cm
20cm
x
Geometry Unit10:Circles
Ms.Talhami 19
13.
AC = ___________(2 dec.)
14.
AC = ___________(2 dec.)
15.
AC = ___________(E)
16.
x = ___________(E)
17.
x = ___________(E)
18.
x = ___________(E)
19.
AB = ___________(E)
20.
x = ___________
21.
x = ___________(E)
Geometry Unit10:Circles
Ms.Talhami 20
CommonTangentLines
TwoTangents
TangenttoaRadius/Diameter
C
B
D
A
mA
B
Geometry Unit10:Circles
Ms.Talhami 21
Practice1.
CB = __________
2.
AC = __________
3.
CB = __________
4.
FA = __________
5.
AB = __________
6.
CB = __________ (E)
7.
CB = ______ (2 dec)
8.
CB = __________
9. Triangle ABC is circumscribed about a circle, and D, E, and F are points of tangency. Let AD = 5, EB = 5, and CF = 10. a) Find the lengths 0f AB, BC, and CA b) Show that ĪABC is isosceles.
10. Triangle ABC is circumscribed about a circle, and D, E, and F are points of tangency. Let AF = 10, CE = 20, and BD = 30. a) Find the lengths 0f AB, BC, and CA b) Show that ĪABC is a right triangle.
10cm
45Ā°
D
C
B
A15cm
8cmC
B
A
55Ā°
9cm
C
B A8cm
6cm
FC
B
A
8cm5cmC
B
A
12cm
30Ā°D C
B
A
4cm2cmG
C
B A
10 3cm
C
B A
Geometry Unit10:Circles
Ms.Talhami 22
11. PQ is tangent to circle O at P, ST is tangent to circle O at S, and OQ intersect circle O at T and R. If OP = 15 and PQ = 20, find: a) OQ____________ b) QS_____________
12. PQ is tangent to circle O at P, ST is tangent to circle O at S, and OQ intersect circle O at T and R. If OQ = 25 and PQ = 24, find PO.
SegmentsFormedINDISEandOUTSIDEaCircle
CA
FE
B
D
E
C
A
D
B
C
D
A B
Geometry Unit10:Circles
Ms.Talhami 23
Examples
Practice
1.
x = _____________
2.
x = _____________
3.
x = _____________
4.
x = _____________
5.
x = _____________
6.
x = _____________
7.
x = _____________
8.
x = __________(2 dec.)
9.
x = ________
10.
x = ________
11.
x = ________
Geometry Unit10:Circles
Ms.Talhami 24
12.
x = ________
13.
x = ________
14.
x = ________
15.
x = ________
16.
x = ________
17.
x = ________
RadianMeasureDoNowAbouthowmanyradiifitaroundacircle?(Hint:thinkbacktoaninscribedhexagonconstruction)
WhatdoesRadianmean?
H
Geometry Unit10:Circles
Ms.Talhami 25
1 Radian
2 Radians
3 Radians
š Radians
4 Radians
5 Radians
6 Radians
šš Radians
ConvertingDegreetoRadianandRadiantoDegreeUnitConversions Proportion
Example1. Express in radian measure an angle of 75o.
2. Find the degree measure of an angle of )
* rad.
r3
r3
r3
r3
r3
r3
r3
r2
r2
r2
r2
r2
r2
r2r1
r1r1
r1
r1
r1
r1
r1rad.r
r
r
r3rad.
r
r
r
rĻrad.
r
r
r
r
r4rad.
r
r
r
r
r
r5rad.
r
r
r
r
r
r
r6rad. r
r
r
r
r
r
r2Ļrad. r
Geometry Unit10:Circles
Ms.Talhami 26
PracticeFind the radian measure or the degree measure of the following: 1. 30o
2. 90o 3. 45o 4. 120o
5. 160o
6. )+
7. ),
8. )-.
9. /)0
10. )/
ArcLength
Example
H
Ļ3
s
5 cm
Ļ2
s
16 cm
Geometry Unit10:Circles
Ms.Talhami 27
Practice1. Central Angle of 30Ā°, radius of 3 cm s = ____________ (E)
2. Central Angle of 90Ā°, radius of 8 cm s = ____________ (E)
3. Central Angle of 72Ā°, radius of 10 cm s = ____________ (E)
4. Central Angle of .4radĻ ,
radius of 12 cm s = ____________ (E)
5. Central Angle of 2 .3radĻ ,
radius of 15 cm s = ____________ (E)
6. Central Angle of 4 .5radĻ ,
radius of 10 cm s = ____________ (E)
ExtraPracticeDeterminethearclengthofthefollowingcircles.1.
2.
3.
4.
Geometry Unit10:Circles
Ms.Talhami 28
AreaofaSector
Practice1. r = 10 cm, Ę = 2 rad.
Area = ________________
2. A = 6Ļ cm2, Ę = 3Ļ rad.
r = ________________
3. r = 3 cm, Ę = 300Ā°
Area = ________________
ExtraPracticeDeterminetheareaofthefollowingsectors.1.
2.
3.
4.
60Ā°5cm
3Ļ rad.
5cm
r
Īø
rĪø
rĪø
Geometry Unit10:Circles
Ms.Talhami 29
Proofs
Radius
1) In a circle, a radius perpendicular to a chord bisects the chord and the arc. 2) In a circle, a radius that bisects a chord is perpendicular to the chord.
3) In a circle, the perpendicular bisector of a chord passes through the center of the circle. 4) If a line is tangent to a circle, it is perpendicular to the radius draw to the point of tangency
Chords
5) In a circle, congruent chords are equidistant from the center. 6) In a circle, congruent chords have congruent arcs. 7) In a circle, congruent arcs have congruent chords. 8) In a circle, parallel chords intercept congruent arcs.
9) In a circle, congruent central angles have congruent chords. Tangents 10) Tangents segments to a circle from the same external point are congruent.
Arcs 11) In a circle, congruent central angles have congruent arcs. Angles 12) An inscribed in a semi-circle is a right angle.
13) The opposite angles of a quadrilateral are supplement. Examples1.
Given: Chords AB, CD, AD, and CB Prove: AEā¢EB = CEā¢ED
Statement
Reasons
2.
Given: Diameters AB and CD Prove: AC ā BD
Statement
Reasons
Geometry Unit10:Circles
Ms.Talhami 30
3.
Given: AB ā AC Prove: ĪAOC ā ĪAOB
Statement
Reasons
4.
Given: Tangent AC Prove: ā O ā ā P
Statement
Reasons