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CONFIDENTIAL 1
GeometryGeometryArcs and chordsArcs and chords
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
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.
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
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
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
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
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
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
CONFIDENTIAL 10
Now you try!
Find each measure.
1) mJKL 2) mLJN
2540
PN
M
L
K
J
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
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
CONFIDENTIAL 13
The converses of the parts of previous theorems are also true. For example, with PART A, congruent chords have congruent
central angles
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
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)
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
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
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
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
CONFIDENTIAL 20
Now you try!
4) Find QR to the nearest tenth.
1010
TS
R
Q
P
CONFIDENTIAL 21
Now some problems for you to practice.
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
CONFIDENTIAL 23
Find each measure.
51
F
E
D
C
B
A
2) mDF 3) mDEB
2) 139°3) 270°
CONFIDENTIAL 24
4) mJL 5) mHLK
30
72
LK
J
H
G
CONFIDENTIAL 25
6y
8y - 8
S
RQ
P
6) QPR RPS.Find QR
CONFIDENTIAL 26
- 9X
(45 - 6X)
F
E
DC
BA
7) A B, and CD EF . Find EBF
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
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
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
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
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
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
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
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
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)
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
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
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
CONFIDENTIAL 39
You did a great job today!You did a great job today!
