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Chapter 10
Circles
10.1
Circles and Circumference
Definitions
 A circle is the locus of all
points in a plane equidistant
from a given point called
the center of the circle.
 The circle is named after
the center point and can be
written as OA if A is the
center point.
A
Center - Point A
Parts of Circles
Chord – A segment with
two endpoints on the circle.
B
BC is a chord
Diameter (d) – A chord that
contains the center point.
BD is a diameter
C
A
Radius (r) – A segment that
contains the center point and
D
any point on the circle.
There are an infinite #
AD & AB are radii of each of these parts.
Radii and Diameters
o Since there are an infinite number of points
on a circle, there can be an infinite number
of radii (r), diameters (d) and chords.
o We know that a diameter contains two radii
and since all the points on a circle are
equidistant from the center point, all radii
are congruent.
o The diameters are twice the radius. (d = 2r
or r = ½ d.
Circumference
o The circumference is the distance around the
circle just like the perimeter of any polygon.
o The Greeks noticed that for every circle the
ratio of the Circumference/Diameter was
always the same. They gave that ratio a
special name.
o Pi (π) which can be abbreviated as 3.14 or
22/7. Pi is an irrational number, meaning it is
a non-repeating, non-terminating decimal.
Circumference (Con’t)
o So, π=C/d (Circumference/diameter) we
can solve for C,
o C = πd and since d = 2r
o C = 2πr
o Using these two formulas we can find the
exact circumference of a circle given either
the radius or the diameter.
o Ex. If d = 4, the C = 4π exactly or approx
12.57
10.2 Angles and Arcs
Angles of Circles
o There are four different types of angles
associated with circles:
o Central Angles
o Inscribed Angles
o Interior Angles
o Exterior Angles
o Central Angle – An angle with the vertex on
the center of the circle and the sides are
radii.
Central Angle
Central Angle – An angle
where the vertex is the
center of the circle and the
sides are radii.
CAB is a central angle
Sum of Central Angles –
The sum of all central
angles equals 360°
B
C
m CAB  m BAD  m DAC  360
A
D
Arcs
o An Arc is a portion of a circle
o There are three types of arcs in a circle:
o Minor Arc
o Major Arc
o Semicircle
o One central angle divides a circle into two
arcs, one minor and one major.
o The sum of all minor arcs equals 360°
Arcs (Con’t)
<CAB is a Central Angle
CB is a minor arc
mCB  m CAB
The remaining arc is the
major arc.
BDC is a major arc
mCB  mBDC  360
B
C
A
D
Semicircle
BC is a diameter
CB is a semicircle
B
CDB is a semicircle
A
mCB  mCDB  180
D
C
Arc Addition Postulate
B
<CAB and <DAC are
adjacent, central angles.
The AAP says:
m<CAB + m<DAC = m<DAB
Since <CAB, <DAC and
<DAB are central angles, the
measurement of the
intercepted arcs are equal to
the angle.
C
A
D
mCB  mDC  mDB
Congruent Arcs
If <CAB and <DAC are
congruent angles, then the
arcs are congruent.
Def of Congruent angles says:
m<CAB = m<DAC
B
C
Def of Central angles says the D
measurement of the arc equals
the measurement of the central
angle.
A
BC  CD
Arc Length
The length of the arc is a
portion of the circumference.
It is not the measurement of
the arc which is measured in
degrees.
B
C
It is the ratio of the measurement
of the arc over 360 times the
circumference.
A
mBC
length BC 
*2 r
360
Example
mBC
BC 
*2 r
360
30
BC 
* 2 (6)
360
1
BC  *12  
12
B
C
A
Find the exact
length of arc BC if
m<BAC is 30° and r
=6
10.3 Arcs and Chords
Theorem 10.2
o In a circle or in congruent circles, if two
chords are congruent, then the arcs of the
chords are congruent.
o In a circle or in congruent circles, if two
arcs are congruent, then the chords that
make those arcs are congruent.
A
B
Proof of Thrm 10.2
Given : AB  CD
A
Draw radii EA, EB ,
EC and ED
EA  EB  EC  ED
B
b / c all radii are 
E
ECD  EAB by SSS
ECD  EAB
C
D
by CPCTC
CD  AB
Circles and Polygons
o We can discuss how circles and polygons
are drawn together.
o Circumscribed – If a circle is outside the
polygon then the circle is said to be
circumscribed about the polygon.
o Inscribed – If a polygon is inside the circle
then the polygon is said to be inscribed in a
circle.
Circles and Polygons
B
A
E
C
D
Quad ABCD is
inscribed in Circle E.
Circle E is
Circumscribed about
Quad ABCD.
Same picture –
notice all vertices of
quad are on the
circle.
Thrm 10.3
o In a circle, if a diameter (or radius) is
perpendicular to a chord, then it bisects the
chord and the arc of the chord.
B
A
C
D
E
o Segment AD is congruent to
Segment DC.
o Arc AB is congruent to Arc BC.
o DE is said to be the distance
that chord AC is from the center
of the circle.
Theorem 10.4
o In a circle or congruent circles, two chords
are congruent if they are equidistant from
the center of the circle.
o In a circle or congruent circles, two chords
are equidistant from the center of the circle
if they are congruent.
Proof of Thrm 10.4
Two Congruent Chords.
B
C
F
A
E
Two radii perpendicular to
two congruent chords.
Two congruent radii.
ΔECB congruent ΔEFD
by HL.
Seg CE congruent to
seg EF by CPCTC.
Seg AB & seg DG equidistant from the center.
10.4 Inscribed Angles
Definition
o An Inscribed angle is an angle with a vertex
on the circle and the sides of the angles are
chords.
C
o <CAB is an inscribed angle
E
A
o Notice that the vertex, A is
B
on the circle and the sides
are chords AC and AB.
Measurement of Arc
C
E
A
B
 Unlike the central angle,
the measurement of the arc
is not equal to the meas. of
the central angle.
 The measurement of the
inscribed angle is ½ the meas.
of the intercepted arc.
1
m CAB  mBC
2
Thrm 10.6
o If two inscribed angles intercept the same
or congruent arcs, then the angles are
congruent.
C
C
B
G
<A and <D intercept <A and <D intercept the same
congruent arcs
EtheseFangles
arc
(arc
BC)
so
E
B so
(arc BC & arc GF)
are congruent.
these angles
A
A
D
are congruent.
D
Example:
W
20°
X
3
Y
4
2
40°
5
F
U
Z
1
T
mUZ  108
Answers:
m<1 = 48°
m<2 = 20°
m<3 = 54°
m<4 = 106°
m<5 = 54°
What If?
o What happens if an inscribed angle
intercepts a semicircle?
A
E
B
oAll these angles
are right angles!
C
oThen it is a right angle
What if?
o What if we had a quadrilateral inscribed in a
circle, what can you say about opposite angles
1
of the quad?
m C
2
1
m A  mDCB
2
mDAB  mDCB  360
A
D
E
C
mDAB
B
2m A  2m C  360
m A  m C  180
Opposite angles are supplementary!
10.5 Tangents
Definition
o A tangent is a line that touches a circle
(curve) at exactly one point.
o That point is the “Point of Tangency”.
Tangent Line
Point of Tangency
(P.O.T.)
E
Theorem 10.9
o If a radius is drawn to the P.O.T., then it is
perpendicular to the tangent line.
Line AB is tangent to Circle E at A.
A
AE _|_ AB
B
E
Theorem 10.10
o If a line is perpendicular to a radius of a
circle at its endpoint, then the line is
tangent to the circle.
AE _|_ AB
A
B
E
Line AB is
tangent to Circle
E at A.
Theorem 10.11
o If two segments from the same exterior
point are tangent to the same circle, then
the tangent segments are congruent.
A
AB is congruent to BC
B
E
C
Polygons and Circles
o Remember when a polygon is inscribed in
a circle all the vertices of the polygon were
inscribed angles?
o Here when a polygon is circumscribed about a
circle (or a circle is inscribed in a polygon), then all
segments are tangent segments.
Four P.O.T.
E
Four Tangent Segments
Example
o ΔJKH is circumscribed about OE. Find the
perimeter of the triangle if NK = JL + 29.
16
J
16
L
M
18
E
H
18
P = 2(16 + 18 + 45)
= 158
16 + 29 = 45
N
16 + 29 = 45
K
Common Tangents
o A common tangent is a tangent that
touches two circles at the same time.
Common External Tangent.
Common Internal Tangent
10.6 Secants, Tangents and
Angle Measurement.
Secant Definition
o Secant – A line that crosses through a circle at
two distinct points.
o Secants contain chords.
B
A
AB is a secant
AB is a chord
Interior Angles
oInterior Angle – An angle where the
vertex is in the interior of the circle and
the sides are “portions” of chords.
C
A
B
E
CBD is an
D
interior angle
Exterior Angles
oExterior Angle – An angle where the vertex
is exterior of the circle and the sides are
combinations of secants, tangents or both.
C
CBD is an
A
D
B
E
exterior angle
Interior Angle
C
A
D
B
E

Point B is in the interior
of Circle
<ABC, <CBD, <DBE and
<EBA are all Interior
Angles
The measure of the Interior
angle is equal to ½ the
sum of the two intercepted
arcs.
1
m CBD  mCD  m AE
2

Exterior Angle
Two Secants
C
A
D
B
E
Point B is in the Exterior
of Circle
<B is an Exterior
Angle
The measure of the Exterior
angle is equal to ½ the
difference of the two
intercepted arcs.

1
m B  mCD  m AE
2

Exterior Angle
Secant Tangent
C
D
B
E
Point B is in the Exterior
of Circle
<B is an Exterior
Angle
The measure of the Exterior
angle is equal to ½ the
difference of the two
intercepted arcs.

1
m B  mCD  mCE
2

Exterior Angle
Two Tangents
C
D
B
E
Point B is in the Exterior
of Circle
<B is an Exterior
Angle
The measure of the Exterior
angle is equal to ½ the
difference of the two
intercepted arcs.

1
m B  mCDE  mCE
2

Inscribed Angle
o Another possible inscribed angle could be
where you have a secant and a tangent.
o The m< = ½ the measure of the intercepted
arcs.
B
m<1 = ½ m Arc AB
C
m<2 = ½ m Arc BCA
2
1
A
10.7 Special Segments in a
Circle
Theorem 10.15 (PP)
o If two chords intersect in a circle, then the
products of the measures of the segments
of the chords are equal.
PP = PP
B
A
E
D
(Part)(Part) = (Part)(Part)
(AE)(EC) = (DE)(EB)
C
Confusing Part: PP looks
like Interior Angle problem,
Int <‘s looking for angles,
here looking for segments.
Proof of Thrm 10.15
E
D
A
B
A
C
B b / c?
ADE ~ BCE b / c ?
AE DE

b / c?
EB CE
Draw auxiliary chords
AD and BC.
( AE )(CE )  ( DE )( EB) b / c ?
D
C b/c?
PP = PP
Example
A
D
3
2
5
E
PP = PP
B
(AE)(EC) = (DE)(EB)
x
C
(2)(x) = (5)(3)
x = (5)(3)/2
x = 15/2
Theorem 10.16 (WE=WE)
B
Be careful:
This looks like
Ext <‘s!
E
D
A
C
WE=WE stands for Whole times
Exterior (of one secant) = Whole
times Exterior (of other secant).
(DB)(DE) = (DC)(DA)
Example
B
E
D
A
C
(EB) = x
(DE) = 8
(AC) = 24
(DA) = 10
(DB)(DE) = (DC)(DA) (8x) = (276)
(x+8)(8) = (34)(10)
x = 34.5
(8x+64) = (340)
Theorem 10.17 WE=TT
B
E
D
Be careful:
This looks like
Ext <‘s!
C
WE=TT stands for Whole times
Exterior (of one secant) = Tangent
times Tangent.
(DB)(DE) = (DC)2
Example
B
E
D
C
(EB) = x + 2
(DE) = x
(DC) = x + 4
(x2 - 6x - 16) = 0
(DB)(DE) = (DC)2
(x – 8)(x + 2) = 0
(2x + 2)(x) = (x + 4)2
x
=
8
(2x2 + 2x) = (x2 + 8x + 16)
x = -2
10.8 Equations of Circles
Equations
o Just like you had to memorize the equation
of a line, you have to memorize the
equation of the circle.
o Also, in y=mx + b, the m is the slope and
the b is the y intercept. What is the x and
y?
o The x and the y were the portions of any
point P(x,y) that was on the line.
Equation of a Circle
o To draw a line you need the slope and the y
intercept. What do you need to draw a
circle?
o You need the center point and the radius.
So those are the things you will have in the
equation.
o Equation of a circle:
o (x – h)2 + (y – k)2 = r2
Equation of a Circle
o Equation of a circle:
o (x – h)2 + (y – k)2 = r2
o The h is the x part of the center, the k is
the y part of the center and r is the length
of the radius.
o The center point C(h,k), r is the radius.
o Does the equation of a circle look like
anything you’ve seen before?
d  ( x2  x1 )2  ( y2  y1 )2
Identify the C and r
of a Circle
o (x – 3)2 + (y – 2)2 = 16
o What is the x part of the center?
oh=3
o What is the y part of the center?
ok=2
o What is the center C(h,k)?
o C(3,2)
or=4
9
Sketch (x – 3)2 + (y – 2)2 = 16
8
7
(3, 6)
6
5
r=4
r=4
4
3
(-1, 2)
-6
-4
(7, 2)
2
1
-8
r=4
-2
C(3, 2)
2
4
-1
-2
-3
(3, -2)
6
8
10
Sketch (x – k)2 + (y – h)2 = r2
(h, k + r)
r=r
(h - r, k)
(h + r, k)
C(h, k)
(h, k - r)
Sketch (x – 3)2 + (y + 2)2 = 25
(3, 3)
(-2,-2)
r=5
(8, -2)
C(3, -2)
(3, -7)
Sketch (x – 3)2 + (y + 2)2 = 5
(3, -2 + √5)
r = √5
(3 - √5,-2)
(3 + √5, -2)
C(3, -2)
(3, -2 -√5)