CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.
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Transcript of CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.
![Page 1: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/1.jpg)
CONFIDENTIAL 1
GeometryGeometryArcs and chordsArcs and chords
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CONFIDENTIAL 2
Warm UpWarm Up
In the figure ,QP are QM tangent to N . Find each measure.
1) mNMQ 2) MQ
2x
4x - 9 108 N
M
Q
P
1) 90°2) 9
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CONFIDENTIAL 3
Arcs and ChordsArcs and Chords
A center angle is an angle whose vertex isthe center of a circle. An arc is an unbroken part of a circle consisting of two points calledthe end points and all the points on the circle
between them.
![Page 4: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/4.jpg)
CONFIDENTIAL 4
Arcs and Their MeasureArcs and Their Measure
ARC MEASURE DIAGRAM
A minor arc is an arcwhose points are on or in the interior of a
center angle.
The measure of a minor arcis equal to the measure of
its center angle.
A major arc is an arcwhose points are on or in the exterior of a
center angle.
The measure of a major arcis equal to minus the
measure of its center angle.
If the endpoints of an arc lie on a diameter,
the arc is a semicircle
The measure of a semicircleis equal to
mAC = mABC = x
360
mADC = 360 - mABC = 360 - x
180
mEFG = 180
A
BC
x
A
BC
x
G
F
E
![Page 5: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/5.jpg)
CONFIDENTIAL 5
The circle graph shows the types of music sold during one week at a music store.
Find
mBC
A
B
CDE
FG
H
Other 26%
M
Rock 25%
Rap 13%
Country 10%Pop 13%
Jazz 13%
Classical 3%
R & B 11%
mBC = mBMC m of arc = m of central mBMC = 0.13(360 ) central is 13% =46.8 of the
![Page 6: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/6.jpg)
CONFIDENTIAL 6
Now you try
Use the graph to find each of the following.
Classical 3%
B
CDE
FG
H
Other 26%
M
Rock 25%
Rap 13%
Country 10%Pop 13%
Jazz 13%
R & B 11%
1) mFMC 2) mAHB 3) mEMD
1) 108°2) 270°3) 36°
![Page 7: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/7.jpg)
CONFIDENTIAL 7
Adjacent arcs are arcs of the same circle that intersect at exactly one point. and are adjacent arcs.RS ST
R
S
T
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CONFIDENTIAL 8
R
S
T
Arc Addition Postulate
The measures of an arc formed by two adjacent arcs is
the sum of the measures of the two arcs.
mABC = mAB + mBC
![Page 9: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/9.jpg)
CONFIDENTIAL 9
Using the Arc Addition Postulate
Find mCDE
mCD = 90 mCFD = 90 mDFE = 18 vert.s Thm.
mDE = 18 mDFE = 18
mCE = mCD + mDE Arc Add.Post = 90 + 18 = 108 Substitute and simlify
18
E
D
C
B
A
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CONFIDENTIAL 10
Now you try!
Find each measure.
1) mJKL 2) mLJN
2540
PN
M
L
K
J
1) 140°2) 250°
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CONFIDENTIAL 11
With in a circle or congruent circles, congruent arcs are
two arcs that have the same measure. In the figure, ST UV
x
x
V
U
T
S
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CONFIDENTIAL 12
THEOREM A HYPOTHESIS CONCLUSION
In a circle or congruent circles
1) Congruent central angles have
congruent chords
2) Congruent chords have congruent arcs
3) Congruent arcs havecongruent central
angles.
E
D C
B
A
E
D C
B
A
E
D C
B
A
EAD BAC
ED BC
ED BC
DE BC
DE BC
DAE DAC
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CONFIDENTIAL 13
The converses of the parts of previous theorems are also true. For example, with PART A, congruent chords have congruent
central angles
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CONFIDENTIAL 14
E
D C
B
A
PART A
Proof:
Given: BAC DAEProve: BC DE
Statements Reasons1) BAC DAE
2) AB AD ,AC AE
3) BAC DAE
4) BE DE
1) Given
2) All radii of a are
3) SAS Steps 2,1
4) CPCTC
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CONFIDENTIAL 15
Applying Congruent Angles, Arcs, and Chords
Find each measure.
RS TU. Find mRS.
RS TU chords have arcs.
mRS = mTU Def. of arcs
3x = 2x+27 Substitute the given measures.
x = 27 Substract 2x from both sides.
mRS = 3(27) Substitute 27 for x.
= 81 Simplify.
(2x + 27)
3x
U
T
S
R
1)
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CONFIDENTIAL 16
B E,and AC DF. Find mDEF.
ABC DEF arcs have central s
mABC = mDEF Def. of s
5y + 5 = 7y - 43 Substitute the given measures. 5 = 2y - 43 substract 5y from both sides. 48 = 2y Add 43 to both sides. 24 = y Divide both sides by 2.
mDEF = 7(24) - 43 Substitute 24 for y =125 Simplify.
2)
(7y - 43)
(5y + 5)
FE
D
CB
A
![Page 17: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/17.jpg)
CONFIDENTIAL 17
Now you try!
Find each measure
PT bisects RPS . Find RT1)
20 - 4x
6x
R
T
S
P
2) A B, and CD EF.
Find mCD.
(30y - 20)
25yFE
D
C
BA
1) 122) 100°
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CONFIDENTIAL 18
THEOREM B HYPOTHESIS CONCLUSION1) In a circle, if a radius
(or diameter) isperpendicular toa chord, then it
bisects the chordand its arc
2) In a circle, theperpendicular
bisector of a chordis a radius (or diameter).
CD EF
E F
D
C
JK is bisector of GH
G
K
J
H
A
CD bisects EF and EF
JK is a diameter of A
![Page 19: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/19.jpg)
CONFIDENTIAL 19
Using Radii and Chords
Find BD.
Step 1 Draw radius AD. AD = 5 Radii of a are .
Step 2 Use the Pythagorean Theorem. CD2 + AC2 = AD2
CD2 + 32 = 52 Substitute 3 for AC and 5 for AD. CD2 = 16 Substract 32 from both sides.
CD = 4 Take the square root of both sides.
Step 3 Find BD. BD = 2(4) = 8 AE BD , so AE bisects BD.
32
E
DCB
A
![Page 20: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/20.jpg)
CONFIDENTIAL 20
Now you try!
4) Find QR to the nearest tenth.
1010
TS
R
Q
P
4) 46.6
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CONFIDENTIAL 21
Now some problems for you to practice.
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CONFIDENTIAL 22
ASSESSMENT
1) The circle graph shows how a typical household spends money on energy. Find each of the following.
6) m UPT5) m RQ4) mUT
3) m SAQ2) m VAU1) m PAQ
1) 198°2) 19.8°3) 61.2°4) 365) 39.66) 324
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CONFIDENTIAL 23
Find each measure.
51
F
E
D
C
B
A
2) mDF 3) mDEB
2) 139°3) 270°
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CONFIDENTIAL 24
4) mJL 5) mHLK
30
72
LK
J
H
G
4) 108°5) 210°
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CONFIDENTIAL 25
6y
8y - 8
S
RQ
P
6) QPR RPS.Find QR
6) 24
![Page 26: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/26.jpg)
CONFIDENTIAL 26
- 9X
(45 - 6X)
F
E
DC
BA
7) A B, and CD EF . Find EBF
7) 135°
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CONFIDENTIAL 27
Arcs and Their MeasureArcs and Their Measure
ARC MEASURE DIAGRAM
A minor arc is an arcwhose points are on or in the interior of a
center angle.
The measure of a minor arcis equal to the measure of
its center angle.
A major arc is an arcwhose points are on or in the exterior of a
center angle.
The measure of a major arcis equal to minus the
measure of its center angle.
If the endpoints of an arc lie on a diameter,
the arc is a semicircle
The measure of a semicircleis equal to
mAC = mABC = x
360
mADC = 360 - mABC = 360 - x
180
mEFG = 180
A
BC
x
A
BC
x
G
F
E
REVIEWArcs and ChordsArcs and Chords
![Page 28: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/28.jpg)
CONFIDENTIAL 28
The circle graph shows the types of music sold during one week at a music store.
Find
mBC
A
B
CDE
FG
H
Other 26%
M
Rock 25%
Rap 13%
Country 10%Pop 13%
Jazz 13%
Classical 3%
R & B 11%
mBC = mBMC m of arc = m of central mBMC = 0.13(360 ) central is 13% =46.8 of the
![Page 29: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/29.jpg)
CONFIDENTIAL 29
Adjacent arcs are arcs of the same circle that intersect at exactly one point. and are adjacent arcs.RS ST
R
S
T
![Page 30: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/30.jpg)
CONFIDENTIAL 30
R
S
T
Arc Addition Postulate
The measures of an arc formed by two adjacent arcs is
the sum of the measures of the two arcs.
mABC = mAB + mBC
![Page 31: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/31.jpg)
CONFIDENTIAL 31
Using the Arc Addition Postulate
Find mCDE
mCD = 90 mCFD = 90 mDFE = 18 vert.s Thm.
mDE = 18 mDFE = 18
mCE = mCD + mDE Arc Add.Post = 90 + 18 = 108 Substitute and simlify
18
E
D
C
B
A
![Page 32: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/32.jpg)
CONFIDENTIAL 32
With in a circle or congruent circles, congruent arcs are
two arcs that have the same measure. In the figure, ST UV
x
x
V
U
T
S
![Page 33: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/33.jpg)
CONFIDENTIAL 33
THEOREM A HYPOTHESIS CONCLUSION
In a circle or congruent circles
1) Congruent central angles have
congruent chords
2) Congruent chords have congruent arcs
3) Congruent arcs havecongruent central
angles.
E
D C
B
A
E
D C
B
A
E
D C
B
A
EAD BAC
ED BC
ED BC
DE BC
DE BC
DAE DAC
![Page 34: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/34.jpg)
CONFIDENTIAL 34
E
D C
B
A
PART A
Proof:
Given: BAC DAEProve: BC DE
Statements Reasons1) BAC DAE
2) AB AD ,AC AE
3) BAC DAE
4) BE DE
1) Given
2) All radii of a are
3) SAS Steps 2,1
4) CPCTC
![Page 35: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/35.jpg)
CONFIDENTIAL 35
Applying Congruent Angles, Arcs, and Chords
Find each measure.
RS TU. Find mRS.
RS TU chords have arcs.
mRS = mTU Def. of arcs
3x = 2x+27 Substitute the given measures.
x = 27 Substract 2x from both sides.
mRS = 3(27) Substitute 27 for x.
= 81 Simplify.
(2x + 27)
3x
U
T
S
R
1)
![Page 36: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/36.jpg)
CONFIDENTIAL 36
B E,and AC DF. Find mDEF.
ABC DEF arcs have central s
mABC = mDEF Def. of s
5y + 5 = 7y - 43 Substitute the given measures. 5 = 2y - 43 substract 5y from both sides. 48 = 2y Add 43 to both sides. 24 = y Divide both sides by 2.
mDEF = 7(24) - 43 Substitute 24 for y =125 Simplify.
2)
(7y - 43)
(5y + 5)
FE
D
CB
A
![Page 37: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/37.jpg)
CONFIDENTIAL 37
THEOREM B HYPOTHESIS CONCLUSION1) In a circle, if a radius
(or diameter) isperpendicular toa chord, then it
bisects the chordand its arc
2) In a circle, theperpendicular
bisector of a chordis a radius (or diameter).
CD EF
E F
D
C
JK is bisector of GH
G
K
J
H
A
CD bisects EF and EF
JK is a diameter of A
![Page 38: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/38.jpg)
CONFIDENTIAL 38
Using Radii and Chords
Find BD.
Step 1 Draw radius AD. AD = 5 Radii of a are .
Step 2 Use the Pythagorean Theorem. CD2 + AC2 = AD2
CD2 + 32 = 52 Substitute 3 for AC and 5 for AD. CD2 = 16 Substract 32 from both sides.
CD = 4 Take the square root of both sides.
Step 3 Find BD. BD = 2(4) = 8 AE BD , so AE bisects BD.
32
E
DCB
A
![Page 39: CONFIDENTIAL 1 Geometry Arcs and chords. CONFIDENTIAL 2 Warm Up 1)90° 2)9.](https://reader036.fdocuments.us/reader036/viewer/2022062408/56649e445503460f94b38f54/html5/thumbnails/39.jpg)
CONFIDENTIAL 39
You did a great job today!You did a great job today!