10.1 notes
Monday, February 23, 2009
2:19 PM
Definition:
1. A circle is the set of points in a plane that are a given distance
from a given point in the plane. The given point is the center of
the circle, and the given distance is the radius.
B
radius
Center
C
D
2. Two or more coplanar circles with the same center are called
concentric circles.
A
3. Two circles are congruent if the have congruent radii.
4. A point is inside(in the interior of the circle) a circle if its
distance from the center is less than the radius.
5. A point is outside(in the exterior of) a circle if its distance from
the center is greater than the radius.
6. A point is on a circle if its distance from the center is equal to
the radius.
D
A
B
C
7. A chord of a circle is a segment joining any two points on the
circle.
8. A diameter of a circle is a chord that passes through the center
of the circle.
Notes Page 1
D
O
A
C
B
Area of a circle = r2
Circumference or perimeter of a circle = d
9. The distance from the center of a circle to a chord is the
measure of the perpendicular segment from the center to the
chord.
D
A
B
C
Theorem 74: If a radius is perpendicular to a chord, then it
bisects the chord.
E
A
D
B
C
Theorem 75: If a radius of a circle bisects a chord that is not
the diameter, then it is perpendicular to that chord.
E
A
D
B
C
Theorem 76: the perpendicular bisector of a chord passes
through the center of the circle.
E
A
D
B
C
Notes Page 2
E
A
D
B
C
Notes Page 3
10.2 notes
Monday, February 23, 2009
8:53 PM
Theorem 77: If two chords of a circle are equidistant from the
center, then they are congruent.
B
C
E
A
F
O
D
Theorem 78: If two chords of a circle are congruent, then they
are equidistant to the center of the circle.
Example 1:
Given: circle O, AB CD
OP = 12x - 5
OQ = 4x + 19
Find: OP
B
C
P
A
O
Q
D
Notes Page 4
Example 2:
Given: Circle O
PQ  TS
Prove: RQ  RS
P
Q
R
O
S
T
Notes Page 5
10.3 notes
Tuesday, February 24, 2009
10:50 AM
Definitions:
1. A central angle is an angle whose vertex is at the center of the
circle.
A
B
C
2. A minor arc < 180
3. major arc > 180
4. A semicircle = 180
X
B
A
5. The measure of a minor arc is the same as the measure of the
central angle.
6. The measure of the major arc is 360 minus the measure of the
minor arc with the same endpoints.
Notes Page 6
minor arc with the same endpoints.
X
B
40
A
Theorem 79: If two central angles of a circle (or congruent
circles) are congruent, then their intercepted arcs are
congruent.
Theorem 80: If two arcs of a circle are congruent, then the
corresponding chords are congruent.
A
D
O
B
C
Theorem 81: If two central angles of a circle are congruent,
then the corresponding chords are congruent.
Theorem 82: If two chords of a circle are congruent, then the
corresponding central angles are congruent.
A
D
O
B
C
Theorem 83: If two arcs of a circle are congruent, then the
Notes Page 7
Theorem 83: If two arcs of a circle are congruent, then the
corresponding chords are congruent.
Theorem 84: If two chords of a circle are congruent, then the
corresponding arcs are congruent.
A
D
O
B
C
Example 1:
Find the mA
A
110 
B
C
Example 2:
Find the length of a chord that cuts off an arc measuring 120
in a circle with radius of 18.
A
120 
B
Notes Page 8
A
120 
B
C
Notes Page 9
10.4 Notes
Tuesday, February 24, 2009
5:08 PM
Definitions:
1. A secant is a line that intersects a circle at exactly two points.
2. A tangent is a line that intersects a circle at exactly one point.
This point is called the point of tangency.
C secant
B
tangent
C
Point of tangency
Postulate: A tangent line is perpendicular to the radius drawn
at the point of tangency.
C
Point of tangency
D
Theorem 85: If two tangent segments are drawn to a circle
from an external point, then those segments are congruent.
AB  AC
C
D
A
Notes Page 10
C
A
D
B
More definitions:
3. Tangent circles are circles that intersect each other at exactly
one point.
4. Two circles are externally tangent if each circle lies outside the
other.
5. Two circles are internally tangent if one of the tangent circles
lies inside the other.
A
B
C
D
externally tangent
internally tangent
6. XY is a common int ernal tan gent
7. AB is a common external tan gent
Y
X
B
A
Notes Page 11
Examples
1. A circle with radius of 8 cm is externally tangent to a circle with
a radius of 18 cm. Find the length of a common external
tangent.
A
B
2. Each side of quadrilateral ABCD is tangent to the circle.
AB = 10, BC =12, AD = 14
A. CD = ____
B. AB + CD = ____
C. BC + AD = ____
Notes Page 12
B
A
C
D
Notes Page 13
10.5 notes
Friday, February 27, 2009
9:50 AM
Central angle
mABC  m AB
Chord -chord Angle
1
mDEC  m AB  mCD
2

A
A
D
B
B
C
Inscribed angle
1
mQ  mPR
2

C
Tangent-chord angle
1
mT  mST
2



P
S
R
T
Q
Secant-secant angle
1
mP  mCD  m AB
2


D

S
A
P
Tangent-tangent Angle
1
mP  mSXT  mST
2
Notes Page 14


D
S
A
P
X
P
B
C
T
Secant-tangent angle
1
mP  mRT  mQT
2

R
Q
P
T
Examples:
1. mDB  20
mDE  104
Find mC
E
D
C
B
A
2. Find mBEC
A
B
29
E
D
47 Notes Page 15

A
B
29
E
47
D
C
3. Find mB
C
110 
B
A
mBC  ____
4.
m AB  ____
110 
A
C
40
B
Notes Page 16
5. x = _____
y= _____
x
100 
y
20
Notes Page 17
10.6 notes
Tuesday, March 03, 2009
10:18 AM
Theorem 89: If two inscribed or tangent-chord angles intercept
the same arc, they are congruent
C
B
A
D
Theorem 90: If two inscribed or tangent-chord angles intercept
congruent arcs , then the they are congruent.
F
E
A
B
D
C
Theorem 91: An angle inscribed in a semicircle is a right angle.
B
A
C
D
Notes Page 18
B
A
C
D
Theorem 92: The sum of the measures of a tangent-tangent
angle and its minor arc is 180
B
D
x
180-x
A
C
Examples:
1. Find BR
Q
6
A
P
R
AB = 13
B
Notes Page 19
2. A secant and a tangent to a circle intersect a 42 angle. The two
intercepted arcs have measures in a 7:3 ratio. Find the
measure of the third arc.
A
third arc
7x
B
3x
42
C
D
Notes Page 20
10.7 notes
Tuesday, March 03, 2009
11:46 AM
Definitions
1. A polygon is inscribed in a circle if all of its vertices lie on the
circle.
B
C
A
D
2. A polygon is circumscribed about a circle if each of its sides is
tangent to the circle.
B
A
O
C
D
3. The center of the circle circumscribed about a polygon is the
circumcenter of the polygon.
B
R
C
A
D
point R is the circumcenter
Notes Page 21
4. The center of a circle inscribed in a polygon is the incenter of
the polygon.
B
A
O
C
D
Point O is the incenter
Theorem 93: If a quadrilateral is inscribed in a circle, its
opposite angles are supplementary.
A + C = 180
B + D = 180
B
C
R
A
D
Theorem 94: If a parallelogram is inscribed in a circle, it must
be a rectangle.
B
A
O
D
C
Notes Page 22
B
A
O
C
D
Example:
1. Circle P is inscribed in trapezoid WXYZ. The radius of circle P is
5 and YZ = 14. Find the perimeter of trapezoid WXYZ
W
Z
P
YZ = 14
X
Y
Notes Page 23
10.8 Notes
Thursday, March 05, 2009
9:06 AM
Theorem 95: Chord-Chord Power Theorem
a b = c d
d
a
c
b
Theorem 96: Tangent-Secant Theorem:
s2 = r(r+t)
t
r
s
Theorem 97: Secant-Secant Power Theorem:
e(e+f) = g(g+h)
f
e
h
Examples:
3
6
Notes Page 24
g
6
3
2
1.
x
24
2.
3
s
3. AB = ____
A
6
C
3
4
B
4. Tangent segment PT measures 8 cm. The radius of the
circle is 6 cm. Find the distance from the point P to the
center of the circle.
T
P
R
Notes Page 25
10.9 notes
Thursday, March 05, 2009
9:51 AM
Definition: The circumference of a circle is its perimeter.
Postulate: C = d
Theorem 98:
 mPQ 
Length of PQ  
d
 360 


Examples:
1. Find the length of a 40 arc of a circle with an 18 cm radius.
2. Two pulleys are connected by a belt. The radii of the pulleys
are 6cm and 30 cm, and the distance between their centers is
48 cm. Find the total length of belt needed to connect the
pulleys.
Notes Page 26