Geometry Chapter 12
Name ______________
10-6 Circles, Arcs, and Central Angles
1. Definition: A circle is the _____________________________________
__________________________________________
a. Sketch a circle by using many
(maybe 20) points that are all
the same distance from A.
A
b. The given point is called the ___________________ of the circle.
c. The given distance is called the __________________.
2. Definition: A radius of a circle is a segment whose endpoints are the
_______________ of the circle and a ________ ______ ______ ________.
Ex. Draw 2 radii (plural of radius)in
circle B. Label them BP and
BQ.
Note: A circle is named by using its center…
B
B
3. Definition: A diameter is a _______________________________
____________________________.
4. Definition: Concentric circles are circles (1) in the same _____________
and (2) with the same ____________________.
5. Definition: Congruent circles are circles that have __________ ________.
6. Definition: An arc is a part of a circle.
7. There are 3 kinds of arcs:
A. Semicircle- half of a circle
B. Minor arc- less than half of a circle
C. Major arc- more than half of a circle.
8. Names of arcs.
A. Semicircle- 3 or more letters must be used.
Ex 1: In O at right, name the 2 semicircles.
_ADC__and__________.
A
D
O
B
C
B. Minor Arc- 2 letters may be used.
Ex. 1: Name the minor arcs.
__AB__, _____, _____, _____, _____
C. Major arc- 3 or more letters must be used.
Ex 1.: Name four major arcs.
_________________________________________
9. Definition: Central angle- angle whose vertex is the center of the circle.
Ex1.: Name the central . ___________
E
A

O
B
D
C
10. Definition: Measure of an arc
A. The measure of a minor arc equals
the measure of its central .
Ex. 1: If AOB = 40, then AB = _______
O
B
A
B. The measure of a semicircle is 180.
A
C. The measure of a major arc equals
360 - minor arc
B
D
Ex 1.: In
O, AC is a diameter.
If AOB = 35,
AB = _____,
BC = _____,
O
ADCB = _____
C
11. Definition: Congruent arcs- arcs that
A. are in the same circle or  s
and B. have the same measure.
A
Examples: Are AB and CD congruent?
(1)
(2)
A
40
38
(3)
A
B
120
(4)
B
C
B
C
D
_______
O
D
C
_______
A
C
120
D
D
B
_______
_______
10-6 Arc Length
12. Circumference of a circle divided by the diameter of that circle equals .
c

d
or
or
or
C=d
C = 2r   (since diameter = 2  radius)
= 2 r
A.  is a non-ending decimal. It never ends!
B. Several approximations of  are 3.14 and
22
.
7
C. Never replace  with one of its approximations unless the directions tell you
to!
13. The length of an arc of a  is a fractional part of the circumference of that .
Formula for the length of an arc:
central
length of ARC
=
360
2 r
Ex.1: Find the length of AB if mAOB = 60 and the radius of O is 6cm.
A

O
B
10-7 Area of Sectors
14. To find the area of a circle, we use
A =  r2
Ex.1: Find the area of a circle with a diameter of 10 units.
15. Definition: A sector of a  is a region bounded by 2 radii and
an arc of the . (A slice of a round pizza is a sector.)
16. The area of a sector of a  is a fractional part of the area of that .
Formula for the area of a sector:
central  area of a sector

360
 r2
Ex.2: Find the area of the sector AOB. mAOB=30
O
A
Ex.3: Find the area of the shaded region. AB= 4 2
(Hint: Find OB, area of AOB, and area of sector AOB.)
6
B
12-1 Tangent Lines
17. Definition: A tangent to a circle is a _____________ that
______________ the circle at _______________ ______
_____________.
Ex. 1:
Draw a circle A with
tangent TG.
Ex. 2:
Draw all the common tangents for
P and
Q. Give the number of
tangents that are "internal tangents" and the number that are "external
tangents."
Case 1
Case 2
_____ internal
_____ internal
_____ external
_____ external
Case 3
Case 4
_____ internal
_____ internal
_____ external
_____ external
18. Draw 2 circles that are internally tangent
19. Draw 2 circles that are externally tangent
20. Definition: A secant is a ______________ that intersects a ____________
at _____________________________.
Ex. 1: Draw
A with secant GF
12-1 Properties of Tangents
21. Theorem: If a line is tangent to a circle, then that line is perpendicular to
the radius drawn to the point of tangency.
T
If TG is tangent to
O
G
O,
then _____________________.
22. Theorem: In a plane, if a line is perpendicular to a radius at its endpoint
on the circle, then the line is tangent to the circle.
Example:
O
PT is tangent to
Q
P
T
O. OQ = 5, QT = 8. Find:
OT ________ OP _________
PT _________
23. Theorem: If 2 segments are drawn from an exterior point tangent to a
circle, then they are congruent.
Example:
P
1. Draw the 2 tangent
segments from P.
2. Label them PQ and PT
3. If PQ = x2 and
PT = 5x + 6,
O
find x and length PT.
(Show algebra.)
12-2 Chords and Arcs
24. Definition: A chord of a circle is a segment whose endpoints are
_______________________________.
Ex. Draw a chord in
C
that is not a diameter.
Label it GE.
C
25. Theorem:
A. In one circle or in congruent circles, if 2 chords are congruent, then their
arcs are congruent.
B. In one circle or in congruent circles, if 2 arcs are congruent, then their
chords are congruent.
A
B
Ex. 1: In
O, if AB  DC, then __________________.
Ex. 2: In
O, if AB  DC, then __________________.
D O
C
26. Theorem: If a diameter (or radius) is  to a chord, then it
A. bisects that chord
B. bisects the minor arc and major arc.
G
If OA  GT, GT = 16, GT = 80, then
D
O X A
GX = _____, GA = _______, GDT = _______
T
27. Theorems: In a or congruent circles,
A. If 2 chords are congruent, they are the same distance from the center,
B. If 2 chords are the same distance from the center, they are congruent.
X
A
B
O
D
Y
C
Ex.:
1. In
O, if AB  CD, then __________________.
2. In
O, if OX = 3 and OY = 4, then ____ > ____
28. Theorem: In a
or congruent circles, if 2 chords are not congruent, then
the longer chord is ___________________ to the center.
B
X
Ex.:
2.3
O
2.4
A
In
O, if OX = 2.3 and OY = 2.4, then
_____________________________________.
Y
12-3 Inscribed Angles
29. Definition: An inscribed angle is an angle whose vertex is on the circle and
whose sides contain chords of the circle.
Ex. 1: Draw a  with inscribed ’s 1 and 2
30. Theorem: The measure of an inscribed angle is half the
measure of its intercepted arc.
B
A

If BC = 80, then BAC = ____
C
31. Theorem: If 2 inscribed angles of a circle intercept the
same arc, then the 2 angles are congruent.
A
1
B
2
What arc does 1 intercept? _____
If the m2 = 20, then m AB = ____and m1= ______.
32. Theorem: An angle inscribed in a semicircle is a right .
Ex.: Since AB is a diameter, C must equal ________.
A

C
B
33. Theorem: If a quadrilateral is inscribed in a circle, then its opposite
angles are supplementary.
Ex.:
1. Draw a circle with inscribed quadrilateral ABCD. If
a.) A = 50, then C = ____.
b.) B = x2 + 20 and D = 9x – 2, find x and mB.
Show Algebra
12-4 Other Angle Measures
34.
Theorem: An angle formed by a chord and a tangent equals ½ the
intercepted arc.
T
Ex.1: TP is tangent to  O PTQ = ½ TQ
If TQ = 140, then PTQ = _____
O

Q
P
35.
Theorem: If 2 chords intersect inside a circle, the measure of each
angle is ½ the sum of the arcs intercepted by the angle and its vertical
angle.
Ex.:
70
B
C
E
AED = ½ (AD + BC)
50

A
= __________
O
= __________
D
AED = __________
150
AEB = __________
BEC = __________
36. Theorem: An angle formed by either
1. 2 secants, OR
2. 2 tangents, OR
3. 1 secant and 1 tangent
that intersect outside a circle have a measure = ½ the difference of the
2 intercepted arcs.
B
1 = ½ (AD – BC)
A
Ex.1: If AD = 80 and BC = 20, then 1 = ____.
1
C
D
Ex.2:
H
If EFG = 260, then 2 = _____.
2
E
G
F
Ex.3:
T
A
T = 35, HA = 90. What is the measure of MYH?
Show Algebra.
M
H

Y
12-4 cont. Segment Lengths in Circles
37. Theorem: When 2 chords intersect inside a , the product of the parts of
1 chord equals the product of the parts of the other chord.
c
a
b
ab=cd
d
A
D
Ex.1: If AX = 4, XC = 7, and DX = 5, then XB = _______.
Show Algebra. (Note: The ’s are . Corr. sides of  ’s
are proportional.)
5
4
B
7
Ex.2: x = ______
3
x
8-x
4
33. Theorem: If 2 chords of a  are parallel, then the 2 arcs between them
are .
Ex.:
G
E
O
M
If EO  GM, then ______________
C
38. Theorem: When 2 secant segments intersect outside a , the product
of 1 secant and its external part equals the product of the other secant
and its external part.
outside1 ● whole1 = outside2 ● whole2
Ex.:
5
6
x
x = _____
4
39. Theorem: When a secant segment and a tangent segment intersect
outside a , the product of the secant and its external part equals the
tangent squared.
Tangent2= outside ● whole
Ex.:
x = ______
6
x
5
12-5 Circles in the Coordinate Plane
40. Equation of a Circle: An equation of a circle with center (h,k) and radius r is
( x  h)2  ( y  k )2  r 2
Ex.1: Write the standard equation of the circle with center (5,-2) and
radius 7.
Ex.2: With center (0,3) and radius 6
Ex.3: With center (-2,-1) and radius
2
41. Using the Center and a Point on a Circle
Ex.4: What is the standard equation of the circle with center (1,-3) that
passes through the point (2,2)?
42. Using Endpoints of a Diameter
Ex.5: What is the standard equation of the circle with diameter endpoints
(-1,1) and (9,5)?
43. Graphing a Circle Given its Equation
Ex.6: Graph the following circles.
2
2
a. ( x  4)  ( y  2)  25
center:_________ r=______
2
2
b. ( x  2)  ( y  3)  16
center:_________ r=______
44. Finding the Location of a Point
2
2
Ex.7: Given circle ( x  2)  ( y  4)  8 , determine if the given point is
inside, outside, or on the circle.
a. (0,6)
b. (1,-2)
c. (-3,5)