CIRCLES
Circle: the set of all points in a plane that are equidistant from a fixed point, which is
called the center of the circle. A circle is named according to its center, usually an O. The
circle symbol is ________. A circle is __________ degrees.
Ex:
Radius: (plural radii) a line segment from the center of the
circle to any point on the circle. All radii of the same circle
are congruent.
Ex: OA ,
C
B
D
Chord: a line segment whose endpoints are points of the
 circle.
Ex: AB ,
O
E
Diameter: a chord that crosses through the center of the
 circle. Remember d = ___ r.
Ex:
A
Central Angle: an angle whose vertex is the center of the circle.
Ex: AOE ,
Inscribed Angle: an angle whose vertex is on the circle and whose sides are chords of the
 circle.
Ex: BAC ,
Arc: any part of the circle. The arc symbol is ______ and it is written over the letters of
 the arc.
ª
Ex: BC
Types of Arcs
Minor Arc: an arc that is less than 180.
ª
Ex: AE

Major Arc: an arc that is greater
than 180. Whenever an arc is 180 or more you name it
using three letters.
º
Ex: ACD

Semicircle: an arc that is 180.
º
Ex: ABC


Quadrant: an arc that is one-fourth of a circle.
ª
Ex: CE
Intercepted Arc: (an arc intercepted by an angle) if each endpoint of the arc is on a
different ray of the angle and the other points of the arc are in the interior of the angle.
ª is intercepted by COE ,
Ex: CE
Measure of an Arc: is equal to the number of the degrees in the central angle that
intercepts it.

ª 
Ex: mLH
H
I
Using the picture to the right and the fact that mIOJ  37 to find
the degree measure of the following:
1. mLOK =
ª 
2. mLK

3. mJOK =
ª 
4. mJK

5. mHOI =
ª 
6. mHI

ª 
7. mHJ
8. mLOI =
9. mLOJ =


10. mHOJ =
ª 
11. mIK

º 
12. mLKI
º
13. mLJK

º
14. mHJL

• 
15. m LI
J
105 °

O
L
K
Inscribed Angles Measures: is one-half the measure of its intercepted arc.
Ex:
C
B
45 °
90 °
A
 An angle inscribed in a semicircle is a right angle.
Ex:
 Two inscribed angles of a circle that intercept the same arc are congruent.
Ex:
A
B
O
D
C
Find each variable.
1.
A
x°
2.
B
3.
4.
x°
y°
O
B
x°
C
146 °
y°
B
D
33 °
A
A
D
O
C
41 °
109 °
x°
z°
C
Find the missing parts of circle O given the following information: OJ = 15cm,
ª  48 , and mLK
ª  57
mILJ  58, mJK
5. OI =

H
I
º
6. mHJK

7. OH =
8. mIOJ =
O
9. mKOJ =


L
ª 
10. mHI
J
•
11. mIJ
K
ª 
12. mLH
ª 
13. mLJI
Find the missing parts of circle O given the following information: mAOD  78 and
ª  74
mBC
ª 
14. mAD

ª 
15. mAB
16. mABD =
C
B
17. mACD =




18. mBDC =
E
19. mBAC =
O
ª 
20. mBC
ª 
21. mCD
º
22. mBDA

D
A
Tangents
Tangent: a line in the plane of the circle that intersects the circle in exactly one
point.
Ex: Line m is tangent to circle O.
B
O
m
 At a given point on a circle, there is one and only
one tangent to the circle.
 If a line is perpendicular to a radius at its point of
intersection with the circle, the line is tangent to the circle.
 If a line is tangent to a circle, the line is perpendicular to the
radius drawn to the point of tangency.
O
n
Tangent Segment: a segment of a tangent line that has one endpoint at the point of
tangency.
Ex:
R
O
P
Q
 Tangent segments drawn to a circle from an external point are congruent.
Ex: PR  PQ
 If two tangents are drawn to a circle from an external point, the line formed by that
point and the center of the circle bisects the angle formed by the tangents.
Ex: OP bisects RPQ .
1. If PR  8cm and RO  3cm find the following:
R
a. mPQ 
b. mOQ 
O
P
c. mOP 
Q
2. If RPQ  38 find each of the following.
a. mRPO 
b. mQOP 
c. mOQP 
Measures of Central Angles, Chords & Arcs
Two arcs have the same measure if and only if their corresponding chords are congruent.
ª  mVU
ª then ST  VU
Ex: If mST
S
T
ª  mVU
ª
Ex: If ST  VU then mST
V
U
ª  30 , ST = SV = VU, and VU = 5 m find the following:
1. If mTU
a. VS =
b. ST =
S
T
ª 
c. mST
ª 
d. mSV
U
V
I
The measures of two minor arcs are equal if and only if their central
angles are congruent.
ª  mJK
ª , then mHOI  mKOJ
Ex: If mHI
J
K
H
O
ª  mJK
ª
Ex: If mHOI  mKOJ , then mHI
ª  80 and mGH
º  80 , find:
2. If mEF
F
ª 
a. mFG
G
• 
b. mEI
H
E
O
ª 
c. mFH
º
d. mEHG

I
e. mEOI 
f. mFOG 
If a line contains the center of a circle and the midpoint of a chord
then the line is a perpendicular bisector of the chord.
A
M
B
suuur
Ex: If OM goes through the midpoint
of AB and the center of the
suur
circle O, then AM  MB and MO  AB .
3. If mZX  20in , V is the midpoint of WY , and OV  5in find each of the following:
a. YV 
b. OX 
Z
Y
V
c. VZ 
O
d. XV 
e. YW 
W
X
Mixed Practice
1. In circle O, the radius is 15 in and chord AB = 18 in. Find each of the following:
a. mAOB 
A
ª 
b. mAB
º
c. mADB

B
O
2. In circle O, LJ = 82 cm, and the perpendicular distance to
KM from the center O is 9 cm. Find each of the following:
D
a. KN =
b. KM =
L
c. mKON 
d. mKOM 
N
K
M
O
ª 
e. mKL
º 
f. mKM
J
º
g. mKMJ

ª 
h. mJM
3. In circle O, YZ  ZU  UV . Find each of the following:
ª 
a. mZU
º 
b. mUW
Z
U
º
c. mYUW

d. mZXU 
Y
V
O
e. mUOV 
f. mZWV 
X
W