circles - introduction circle – the boundary of a round region in a plane a
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Circles - Introduction
Circle – the boundary of a round region in a plane
A
Circles - Introduction
Circle – the boundary of a round region in a plane
A
- The set of all points in a plane that are given distance from a given point in the plane. P
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P
The center of this circle is the given point in the middle, point P.
Circles are named by their center point. So in this case we have circle P.
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P
The center of this circle is the given point in the middle, point P.
Circles are named by their center point. So in this case we have circle P.
Radius – a line segment from the center out to the edge of the circle
- it measures the distance from the center to any point on the circle
O
PO
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P
The center of this circle is the given point in the middle, point P.
Circles are named by their center point. So in this case we have circle P.
Radius – a line segment from the center out to the edge of the circle
- it measures the distance from the center to any point on the circle
Diameter – a line segment that has endpoints on the circle and goes through the center
O
PO
N
M
MN
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P
The center of this circle is the given point in the middle, point P.
Circles are named by their center point. So in this case we have circle P.
Radius – a line segment from the center out to the edge of the circle
- it measures the distance from the center to any point on the circle
Diameter – a line segment that has endpoints on the circle and goes through the center
- two times larger than the radius
O
PO
N
M
MNrd 2
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P O
RS
N
M
Chord – a line segment that joins any two points on the circle.
S
R
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P O
RS
N
M
Chord – a line segment that joins any two points on the circle.
The interior of the circle are the points contained inside the circle ( blue shading )
The exterior of the circle are the points sitting outside the circle ( gray shading )
S
R
Circles - Introduction
Circle – the boundary of a round region in a plane
A
-The set of all points in a plane that are given distance from a given point in the plane.
-These points form a round line around point P…
P O
RS
N
M
Chord – a line segment that joins any two points on the circle.
The interior of the circle are the points contained inside the circle ( blue shading )
The exterior of the circle are the points sitting outside the circle ( gray shading )
Multiple radii can be drawn from the center.
S
R T
PNPOPTPM ,,,
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
A
P
QM N
a
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
A
P
QM N
a
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
The converse is true as well :
If a line through the center of a circle bisects a chord, it is perpendicular to that chord.
A
P
QM N
a
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half A
P
QM N
a
NQMQ
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
A
P
QM N
a
NQMQ
PQNPQM and
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = ?
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = 10
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = 10
QM = 8 QN = ?
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = 10
QM = 8 QN = 8
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = 10
QM = 8 QN = 8
If MN = 30 QN = ?
Circles - Introduction
Line Chord Theorem :
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
It’s referred to as a perpendicular bisector
It cuts segment MN in half
If we draw to radii, we create two right triangles
These two triangles are congruent.
A
P
QM N
a
NQMQ
PQNPQM and
PQPQ
QMQN
PNPM
EXAMPLE : Fill in the table
PN = 10 PM = 10
QM = 8 QN = 8
If MN = 30 QN = 15
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
Z C
A B
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
SCBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
CBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
3 then ,6 If CSCB
3
CBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
3 then ,6 If CSCB
3
given s which wa4RS
4
CBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
3 then ,6 If CSCB
3
given s which wa4RS
4
CBRS - from above statement
CBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
3 then ,6 If CSCB
3
given s which wa4RS
4
CBRS - from above statement
Using Pythagorean theorem…
5
25
916
34
2
2
222
222
RC
RC
RC
RC
SCRSRC
CBRS
Circles – Introduction
If a line through the center of a circle is perpendicular to a chord, then the line bisects the chord
With this in mind, we can solve problems like this one.
Z C
A B
R. circle ofdiameter theFind
4
6
RS
CB
R
S
Solution : First draw in your radius
3 then ,6 If CSCB
3
given s which wa4RS
4
CBRS - from above statement
Using Pythagorean theorem…
5
25
916
34
2
2
222
222
RC
RC
RC
RC
SCRSRC
Diameter = 10 rd 2
CBRS
Circles – Introduction
Example # 2 :
Diameter of Circle M = 20
Segment RQ = 8
Find MQ
Z
tM
R
Q
SRt S
Circles – Introduction
Example # 2 :
Diameter of Circle M = 20
Segment RQ = 8
Find MQ
Z
S
M
R
Q
102
20
2
diam radius
10
RQ = 8 which was given
8
SRt
t
Solution : If Diameter of M = 20, then MR = 10
Circles – Introduction
Example # 2 :
Diameter of Circle M = 20
Segment RQ = 8
Find MQ
Z
S
M
R
Q
Solution : If Diameter of M = 20, then MR = 10
102
20
2
diam radius
10
RQ = 8 which was given
8
6
36
64100
810
2
2
222
222
MQ
MQ
MQ
MQ
QRMRMQ
Again use Pythagorean theorem…
SRt
t
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
DNQ
B
A
Ct
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
DNQ
B
A
Ct
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
DNQ
B
A
Ct
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
DNQ
B
A
Ct
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
If MQ = 48, then NC = 48.
DNQ
B
A
Ct
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
If MQ = 48, then NC = 48.
That gives us 2 sides of a right triangle. ( ∆NAC )
DNQ
B
A
Ct
N
A
C48
36
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
If MQ = 48, then NC = 48.
That gives us 2 sides of a right triangle. ( ∆NAC )
If we can find NA, we will know ND, because ND is also a radius of circle N.
DNQ
B
A
Ct
N
A
C48
36
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
If MQ = 48, then NC = 48.
That gives us 2 sides of a right triangle. ( ∆NAC )
If we can find NA, we will know ND, because ND is also a radius of circle N.
DNQ
B
A
Ct
N
A
C48
3660
3600
12962304
3648
2
2
222
NA
NA
NA
NA
Circles – Introduction
Example # 3 :
Circle M = Circle N
Line t bisects and is perp.
to SR and AB
AB = SR
SR = 72
MQ = 48
Find ND
S
M
R
Solution : If circle M = circle N, then the parts of these circles are congruent, which would include radius, diameter, and chords.
It was given that AB = SR and SR = 72, so AB = 72.
If AB = 72, AC and CB = 36.
If MQ = 48, then NC = 48.
That gives us 2 sides of a right triangle. ( ∆NAC )
If we can find NA, we will know ND, because ND is also a radius of circle N.
If NA = 60, then ND = 60
DNQ
B
A
Ct
N
A
C48
3660
3600
12962304
3648
2
2
222
NA
NA
NA
NA
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
M
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
M
N
D
C
SR
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
EXAMPLE :
N
D C
A BX
Y
XY bisects and is perpendicular to AB and CD.
AB = 24 and NB = 20, find XY.
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
EXAMPLE :
N
D C
A BX
Y
XY bisects and is perpendicular to AB and CD.
AB = 24 and NB = 20, find XY.
12 then ,24 If XBAB12
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
EXAMPLE :
N
D C
A BX
Y
XY bisects and is perpendicular to AB and CD.
AB = 24 and NB = 20, find XY.
given as which w20
12 then ,24 If
NB
XBAB12
20
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
EXAMPLE :
N
D C
A BX
Y
XY bisects and is perpendicular to AB and CD.
AB = 24 and NB = 20, find XY.
given as which w20
12 then ,24 If
NB
XBAB12
20
16256
144400
1220 22
NX
NX
NX16
Circles – Introduction
Chords that are the same distance from the center of a circle have equal length.
This also applies to congruent circles.
If circle M = circle N,
and if the red distances are equal,
then CD = RS
EXAMPLE :
N
D C
A BX
Y
XY bisects and is perpendicular to AB and CD.
AB = 24 and NB = 20, find XY.
given as which w20
12 then ,24 If
NB
XBAB12
20
16256
144400
1220 22
NX
NX
NX
32162
2
XY
NXXY