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Circles
Chapter 12
Parts of a Circle
A
•
E•
Circle: The set of all points in a plane
•B
that are a given distance from a given
point in that plane.
Center: The middle of the circle – a
M•
diameter
circle is named by its center, the
symbol of a circle looks like - ‫סּ‬M.
•C
•
D
Radius: a segment that has one
endpoint at the center and the other
endpoint on the circle. The radius is
½ the length of the diameter. ALL
RADII ARE CONGRUENT.
More Vocabulary
A
•
Chord: a segment that has its
•B
endpoints on the circle.
Diameter: a chord that passes
E•
through the center of the circle.
The diameter is 2 times the radius.
M•
Circumference: the distance
diameter
•C
•
D
around the circle. To find the
circumference use: C  2r
Arcs: the space on the circle
between the two points on the
circle.
Tangent Lines
Tangent line: A line that
intersects the circle at exactly
one point. AB is a tangent
line to ‫סּ‬T.
Point of Tangency: the
point where the circle and the
tangent line intersect.
•A
•P
T•
Theorem 12-1: If a line is
tangent to a circle, then the line is
perpendicular to a radius drawn to
the point of tangency.
AB  TP
•B
Tangent Lines Continued
•Q
P
•
•
X
Theorem 12-3: Two
•R
segments tangent to a circle from
the same point outside of the
circle are congruent.
PQ  QR
Central Angles
Central Angles: angles whose vertex is
the center of the circle.
•G
D•
Theorem 12-4:
37º
Within a circle or congruent circles:
37º
(1) Congruent central angles have
congruent chords.
•E
F•
DF  DG
(2) Congruent chords have congruent
arcs.
DF  DG
(3) Congruent arcs have congruent
central angles.
GED  DEF
Chords
Theorem 12-5:
Within a circle or congruent circles
(1) Chords equidistant from the
center are congruent.
AB  CD
(2) Congruent chords are
equidistant from the center.
EP  PF
•B
E
•
A•
P•
C•
•D
•F
More About Chords
Theorem 12-6:
In a circle, a diameter that is perpendicular
to a chord bisects the chord and its arc.
UV  VS
•U
UR  RS
Theorem 12-7:
In a circle, a diameter that bisects a Q •
chord (that is not a diameter) is
perpendicular to the chord.
QR  US
Theorem 12-8:
In a circle, the perpendicular bisector of a
chord contains the center of the circle.
•
T
V
•
•S
•R
Inscribed Angles
Inscribed Angle: An angle
whose vertex is on the circle,
and the sides are chords of
the circle. C
Intercepted Arc: an arc of
a circle having endpoints on
the sides of an inscribed
angle. AB
A
•
D•
B
•
•C
Inscribed Angle Theorem
The measure of an inscribed
angle is half the measure of its
intercepted arc.
A ●
90°
1
mB  mAC
2
Corollaries:
1. Two inscribed angles that intercept
the same arc are congruent.
2. An inscribed angle in a semicircle is a
right angle.
3. The opposite angles of a quadrilateral
inscribed in a circle are
1
supplementary.
4
●B
45°
C●
38°
1  380
●
1
m2  90
2
3
●
0
●
2
●
m1  m3  1800
An angle formed by a tangent
line and a chord.
The measure of angle formed by a
tangent line and a chord is half the
measure of the intercepted arc.
B
●
1
mC  mBDC
2
●
D●
130°
65°
●
C
Secant Line
A secant line is a line that intersects a circle
at two points.
Theorem 12-11:
The measure of an angle formed by two lines
that
(1) Intersect inside a circle is half the sum of
the measure of the intercepted arcs.
x°
1
m1   x  y 
2
1
(2) Intersect outside the circle is half the
difference of the measures of the
intercepted arcs.
1
m1 
2
x  y 
y°
●
x°
1
●
●
y°
●
Segment Length Theorems
3. (y + z)y = t2
1. a ● b = c ● d
t
c
a
d
● b
●
2. (w + x)w = (y + z)y
z
x
w
z
y
●
y
Equation of a Circle
An equation of a circle with the center (h, k)
and radius r is (x – h)2 + (y – k)2 = r2.
Example: Center (5, 3) radius 4.
(x – 5)2 + (y – 3)2 = 16