Circles

§ 11.1 Parts of a Circle

§ 11.2 Arcs and Central Angles

§ 11.3 Arcs and Chords

§ 11.4 Inscribed Polygons

§ 11.5 Circumference of a Circle

§ 11.6 Area of a Circle
Parts of a Circle
You will learn to identify and use parts of circles.
1) circle
2) center
3) radius
4) chord
5) diameter
6) concentric
Parts of a Circle
A circle is a special type of geometric figure.
center point
All points on a circle are the same distance from a ___________.
A
O
B
The measure of OA and OB are the same; that is, OA  OB
Parts of a Circle
A circle is the set of all points in a plane that are a given
point in the plane, called the
distance from a given _____
center of the circle.
______
Definition
of a
Circle
P
Note: a circle is named by its center. The circle above is named circle P.
Parts of a Circle
There are three kinds of segments related to circles.
radius is a segment whose endpoints are the center of the circle and a
A ______
point on the circle.
A _____
chord is a segment whose endpoints are on the circle.
diameter is a chord that contains the center
A ________
Parts of a Circle
radius
chord
diameter
R
Segments
of
Circles
K
J
K
K
T
A
KA is a radius
of
K
G
JR is a chord
of
K
GT is a diameter
of
K
chord
From the figures, you can not that the diameter is a special type of _____
that passes through the center.
Parts of a Circle
Use
Q to determine whether each statement is true or false.
AD is a diameter of
Q.
B
False;
Segment AD does not go
through the center Q.
A
Q
AC is a chord of
Q.
C
True;
D
AD is a radius of
False;
Q.
Parts of a Circle
Theorem
11-1
congruent
All radii of a circle are _________.
R
G
PR  PG  PS  PT
P
S
Theorem
11-2
T
The measure of the diameter d of a circle is twice the
measure of the radius r of the circle.
d  2r
or
1
d r
2
Parts of a Circle
B
Find the value of x in
3x – 5 = 2(17)
Q.
17
A
3x – 5 = 34
3x = 39
x = 13
Q
3x-5
C
Parts of a Circle
P has a radius of 5 units, and
T has a radius of 3 units.
If QR = 1, find RT
RT = TQ – QR
A
RT = 3 – 1
P
RT = 2
If QR = 1, find PQ
If QR = 1, find AB
PQ = PR – QR
AB = 2(AP) + 2(BT) – 1
PQ = 5 – 1
AB = 2(5) + 2(3) – 1
PQ = 4
AB = 10 + 6 – 1
AB = 15
Q
R
T
B
Parts of a Circle
P has a radius of 5 units, and
T has a radius of 3 units.
If AR = 2x, find AP in terms of x.
AR = 2(AP)
2x = 2(AP)
x = AP
If TB = 2x, find QB in terms of x.
QB = 2(TB)
QB = 2(2x)
QB = 4x
A
P
Q
R
T
B
Parts of a Circle
Because all circles have the same shape, any two circles are similar.
congruent
However, two circles are congruent if and only if (iff) their radii are _________.
Two circles are concentric if they meet the following three requirements:
 They lie in the same plane.
T
 They have the same center.
 They have radii of different lengths.
R
S
Circle R with radius RT and
circle R with radius RS are
concentric circles.
Arcs and Central Angles
You will learn to identify major arcs, minor arcs, and semicircles
and find the measures of arcs and central angles.
1) central angle
2) arcs
3) minor arc
4) major arc
5) semicircle
6) adjacent arc
Arcs and Central Angles
central angle is formed when the two sides of an angle meet at the center
A ____________
of a circle.
arcs that are
Each side intersects a point on the circle, dividing it into ____
curved lines.
There are three types of arcs:
minor arc is part of the circle in the interior of
A _________
the central angle with measure less than 180°.
major arc is part of the circle in the exterior of
A _________
the central angle.
Semicircles
__________ are congruent arcs whose endpoints
lie on the diameter of the circle.
T
R
central
angle
S
Arcs and Central Angles
minor arc PG
P
Types
of
Arcs
major arc PRG
R
P
semicircle PRT
R
K
K
K
G
m PG  180
P
G
m PRG  180
T
G
m PRT  180
Note that for circle K, two letters are used to name the minor arc, but three letters are
used to name the major arc and semicircle. These letters for naming arcs help us
trace the set of points in the arc. In this way, there is no confusion about which arc
is being considered.
Arcs and Central Angles
Depending on the central angle, each type of arc is measured in the
following way.
Definition
of Arc
Measure
1) The degree measure of a minor arc is the degree measure
of its central angle.
2) The degree measure of a major arc is 360 minus the degree
measure of its central angle.
3) The degree measure of a semicircle is 180.
Arcs and Central Angles
In
P, find the following measures:
m MA = APM
M
A
46°
m MA = 46°
P
H
APT
 m AT
APT = 80°
m THM = 360° – (MPA + APT)
m THM = 360° – (46° + 80°)
m THM = 360° – (126°)
m THM = 234°
80°
T
Arcs and Central Angles
In
M
adjacent arcs.
P, AM and AT are examples of ________
46°
Adjacent arcs have exactly one point in common.
For AM and AT, the common point is __.
A
A
P
H
80°
T
Adjacent arcs can also be added.
The sum of the measures of two adjacent arcs is the measure
of the arc formed by the adjacent arcs.
Postulate
11-1
Arc
Addition
Postulate
P
Q
C
If Q is a point of PR, then
mPQ
R
+ mQR = mPQR
Arcs and Central Angles
In
P, RT is a diameter.
Find mQT.
mQT + mQR = mTQR
mQT + 75° = 180°
mQT = 105°
R
75°
S
65°
Q
P
Find mSTQ.
mSTQ + mQR + mRS = 360°
mSTQ + 75° + 65° = 360°
mSTQ + 140° = 360°
mSTQ = 220°
T
Arcs and Central Angles
Suppose there are two concentric circles with ASD forming two minor arcs,
BC and AD.
A
B
Are the two arcs congruent?
60°
D
S
C
Although BC and AD each measure 60°, they are not congruent.
The arcs are in circles with different radii, so they have different lengths.
However, in a circle, or in congruent circles, two arcs are congruent if they
have the same measure.
Arcs and Central Angles
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding central angles are
congruent.
Theorem
11-3
Y
W
WX  YZ
60°
iff
mWQX = mYQZ
Z
Q
60°
X
Arcs and Central Angles
In
M, WS and RT are diameters, mWMT = 125, mRK = 14.
Find mRS.
WMT  RMS
WMT = RMS
Vertical angles are congruent
Definition of congruent angles
R
W
K
mWT = mRS
Theorem 11-3
M
125 = mRS
Substitution
T
Find mKS.
Find mTS.
KS + RK = RS
TS + RS = 180
KS + 14 = 125
TS + 125 = 180
KS = 111
TS = 55
S
Arcs and Central Angles
Twenty-two percent of all teens ages 12 through 17 work either full or part-time.
The circle graph shows the number of hours they work per week.
Find the measure of each central angle.
Teens at Work
1 to 5
20%
1 – 5: =20%
72 of 360
6 – 10: =23%
83 of 360
11 – 20:
30 or More
10%
6 to 10
23%
=33%
119of 360
21 – 30: =14%
50 of 360
30 or More: =10%
36 of 360
21 to 30
14%
11 to 20
33%
Source: ICRs TeenEXCEL survey for Merrill Lynch
Arcs and Chords
You will learn to identify and use the relationships among
arcs, chords, and diameters.
Nothing New!
Arcs and Chords
In circle P, each chord joins two points on a circle.
Between the two points, an arc forms along the circle.
By Theorem 11-3, AD and BC are congruent
because their corresponding central angles are
vertical angles and therefore congruent.
_____________,
By the SAS Theorem, it could be shown that
ΔAPD  ΔCPB.
congruent
Therefore, AD and BC are _________.
C
A
P
D
B
The following theorem describes the relationship between two congruent
minor arcs and their corresponding chords.
S
Arcs and Chords
In a circle or in congruent circles, two minor arcs are congruent
if and only if (iff) their corresponding ______
chords are congruent.
Theorem
11-4
A
AD  BC
iff
D
B
AD  BC
C
Arcs and Chords
The vertices of isosceles triangle ABC are located on
R.
If BA  AC, identify all congruent arcs.
A
BA  AC
R
C
B
Arcs and Chords
G
Step 1) Use a compass to draw circle on a
piece of patty paper. Label the
center P. Draw a chord that is not
a diameter. Label it EF.
Step 2) Fold the paper through P so that
E and F coincide. Label this fold
as diameter GH.
F
E
P
H
Q1: When the paper is folded, how do the lengths of EG and FG compare?
EG  FG
Q2: When the paper is folded, how do the lengths of EH and FH compare?
EG  FG
Q3: What is the relationship between diameter GH and chord EF?
They appear to be perpendicular.
Arcs and Chords
In a circle, a diameter bisects a chord and its arc if and only if
(iff) it is perpendicular to the chord.
A
AR  BR and AD  BD
iff
Theorem
11-5
C
P
CD
AB
R
D
B
Like an angle, an arc can be bisected.
Arcs and Chords
Find the measure of AB in
AB  2DB
AB  27 
K.
Theorem 11-5
Substitution
AB  14
A
C
D
K
7
B
Arcs and Chords
Find the measure of KM in
KM 2  KN 2  MN2
KM 2  62  82
KM 2  36  64
KM 2  100
KM 2
K if ML = 16.
Pythagorean Theorem
Given; Theorem 11-5
L
K
 100
KM  10
6
N
K
M
Inscribed Polygons
You will learn to inscribe regular polygons in circles and explore
the relationship between the length of a chord and its distance
from the center of the circle.
1) circumscribed
2) inscribed
Inscribed Polygons
circumscribed
When the table’s top is open, its circular top is said to be ____________
about the square.
inscribed in the circle.
We also say that the square is ________
Definition
of
Inscribed
Polygon
A polygon is inscribed in a circle if and only if every vertex of the
polygon lies on the circle.
Inscribed Polygons
Some regular polygons can be
constructed by inscribing them in circles.
A
B
Inscribe a regular hexagon, labeling
the vertices, A, B, C, D, E, and F.
Construct a perpendicular segment
from the center to each chord.
From our study of “regular polygons,”
we know that the chords
AB, BC, CD, DE, and EF are
congruent
_________
C
F
P
D
E
From the same study, we also know that all of the perpendicular segments,
congruent
called apothems
________, are _________.
The chords
are congruent
because
the distances
from
the center
the
Make
a conjecture
about the
relationship
between the
measure
of theof
chords
circle
congruent.
and
theare
distance
from the chords to the center.
Inscribed Polygons
In a circle or in congruent circles, two chords are congruent
equidistant from the center.
if and only if they are __________
B
Theorem
11-6
M
AD  BC
iff
P
LP  PM
A
L
C
D
Inscribed Polygons
R
In circle O, O is the midpoint of AB.
S
If CR = -3x + 56 and ST = 4x,
find x
A
O
OA OB
definition of midpoint
CR  ST
Theorem 11-6
3x  56  4x
56  7x
8 x
substitution
B
C
T
Circumference of a Circle
You will learn to solve problems involving circumferences of
cirlces.
1) circumference
2) pi (π)
Circumference of a Circle
An in-line skate advertises “80-mm clear wheels.”
The description “80-mm” refers to the diameter of the skates’ wheels.
As the wheels of an in-line skate complete one revolution, the
distance traveled is the same as the circumference of the wheel.
Just as the perimeter of a polygon is the distance around the polygon,
the circumference of a circle is the ______________________.
distance around the circle
Circumference of a Circle
On a sheet of paper, create a table similar to the one below:
Circumference
Example Data
271
Diameter
Ratio:
C/D
Result
86
271 ÷ 86
3.151163
Go to this website to Collect Data from various Circles.
Go to this website to Analyze your Data.
Circumference of a Circle
In the previous activity, the ratio of the circumference C of a circle to its
diameter d appears to be a fixed number slightly greater than 3, regardless
of the size of the circle.
The ratio of the circumference of a circle to its diameter is always fixed and
pi or __.
π
equals an irrational number called __
C = __,
Thus, ____
π or C  d .
d
Since d  2r, the relationship can also be writtenas C  2r.
Circumference of a Circle
If a circle has a circumference of C units and a radius of r
d
units, then C = ____
2r or C = ___
Theorem
11-7
Circumference
of a Circle
d
r
C
Circumference of a Circle
Circumference of a Circle
Area of a Circle
You will learn to solve problems involving areas and sectors
of circles.
1) sector
Area of a Circle
The space enclosed inside a circle is its area.
By slicing a circle into equal pie-shaped pieces as shown below, you can
rearrange the pieces into an approximate rectangle.
Note that the length along the top and bottom of this rectangle equals the
circumference of the circle, ____.
_____________
2r
So, each “length” of this approximate rectangle is half the circumference,
or __
r
Area of a Circle
The “width” of the approximate rectangle is the radius r of the circle.
Recall that the area of a rectangle is the product of its length and width.
Therefore, the area of this approximate rectangle is (π r)r or
2

r
___.
Area of a Circle
If a circle has an area of A square units and a radius of
2

r
r units, then A = ___
Theorem
11-8
Area
of a Circle
r
A  r 2
Area of a Circle
Find the area of the circle whose circumference is 6.28 meters.
Round to the nearest hundredth.
A  r 2
A   (1)2
A
A  3.14 m 2
You could calculate the
Use your knowledge
area if you only knew
of circumference.
the radius.
Solve for radius, r.
Any ideas?
C  2r
C
r
2
6.28
r
2
1 r
Area of a Circle
Theorem 11-9
Area of a
Sector of a
Circle
If a sector of a circle has an area of A square units, a
central angle measurement of N degrees, and a radius of
r units, then
 
N
A
 r2
360
Area of a Circle