Transcript Document

Chapter 10 Section 3
 What
is a central angle?
 What is a major arc?
central angle
A
How do you find the measure of a
major arc?
 How do you name a major arc?

 What
is a minor arc?
How do you find the measure of a
minor arc?
 How do you name a minor arc?
major
arc
minor
arc
P

 What
is a semi-circle?
 Two arcs are congruent
when…
B
C
A

point H is called
the midpoint ofCHB
if C H  HB . Any
line, segment, or
ray that contains H
bisects CHB .
 

 What



is a chord?
Any segment with
endpoints that are on
the circle
The endpoints of a
chord are also the
endpoints of an arc.
AB is a chord, while
AB is formed by the
same endpoints.


A
In the same circle, or in
congruent circles, two
minor arcs are
congruent if and only if
their corresponding
chords are congruent.

AB  BC if and only if
AB  BC
C
B

If a diameter (or radius)
of a circle is
perpendicular to a chord,
then the diameter bisects
the chord and its arc.
F
 
DE  EF ,
E
DG  GF

From this, if one chord is a
perpendicular bisector of
another chord, then the first
chord is a diameter.
G
D
(x + 40)°
D
 You
can use
Theorem 10.2 to
find m AD .



 
C
A
• Because AD  DC,
and AD  DC . So,
m AD = m DC
2x = x + 40
x = 40
2x°
B
Substitute
Subtract x from each
side.
 Step
1: Draw any
two chords that
are not parallel to
each other.
 Step
2: Draw the
perpendicular
bisector of each
chord. These are
the diameters.
 Step
3: The
perpendicular
bisectors intersect
at the circle’s
center.
center
 In
the same circle, or in
congruent circles, two
chords are congruent if and
only if they are equidistant
from the center.

C
G
D
E
(recall distance is measured by
a perpendicular line)
 CD if and only if EF 
EG.
 AB
B
F
A
AB = 8; DE = 8, and
CD = 5. Find CF.
A
8 F
B
C
E
5
8
G
D
Because AB and DE
are congruent
chords, they are
equidistant from the
center. So CF  CG.
To find CG, first find
DG.
CG  DE, so CG
bisects DE. Because
DE = 8, DG = 82 =4.
A
8 F
B
C
E
5
8
G
D
Then use DG to find
CG.
DG = 4 and CD = 5, so
∆CGD is a 3-4-5
right triangle. 3
So CG = 3. Finally,
use CG to find CF.
Because CF  CG,
CF = CG = 3
A
8 F
B
C
E
5
8
G
D
 Inscribed



polygons-
All vertices lie on the
circle
Contained within the
circle
Chords of adjacent
arcs form an inscribed
polygon
 Circumscribed
poylgons

Contains all the
vertices of another
polygon
Polygon ABCDEFGH is
circumscribed about
the circle
Chapter 10 Section 4
An inscribed angle is
an angle whose
vertex is on a circle
and whose sides
contain chords of
the circle.
inscribed angle
 The arc that lies in
the interior of an
inscribed angle and
has endpoints on the
angle is called the
intercepted arc of
the angle.

intercepted arc
an angle is
inscribed in a circle,
then its measure is
one half the
measure of its
intercepted arc.
mADB = ½m
A
 If

AB
C
D
B
 Find
the measure of
the blue arc.
S
R

•mQTS = 2mQRS
=
•2(90°) = 180°
T
Q
 Find
the measure of
the blue arc.

•m ZWX= 2mZYX
= •2(115°) = 230°
W
Z
•115°
Y
X
 Find
the measure of
the blue angle.


•mNMP = ½ NP
m•½ (100°) = 50°
N
•100°
M
P

Find mACB, mADB,
and mAEB.
A
•60
The measure of each
angle is half the
measure of AB
E
m AB = 60°, so the
measure of each angle
is 30°


B
D
C
 If
two inscribed
angles of a circle
intercept the same
arc, then the
angles are
congruent.
 C  D
A
D
B
C
 It
is given that mE
= 75°. What is mF?
 E

and F both
intercept GH , so E
 F. So, mF =
mE = 75°
G
E
•75
°
F
H

Theater Design.
When you go to the
movies, you want to
be close to the movie
screen, but you don’t
want to have to move
your eyes too much to
see the edges of the
picture.
 If
E and G are the
ends of the screen
and you are at F,
mEFG is called
your viewing
angle.
 You
decide that
the middle of the
sixth row has the
best viewing
angle. If
someone else is
sitting there,
where else can
you sit to have
the same viewing
angle?
 Solution:
Draw
the circle that is
determined by
the endpoints of
the screen and
the sixth row
center seat. Any
other location on
the circle will
have the same
viewing angle.
 If
all of the vertices
of a polygon lie on a
circle, the polygon is
inscribed in the circle
and the circle is
circumscribed about
the polygon.
 The polygon is an
inscribed polygon and
the circle is a
circumscribed circle.



Theorem 10.7- If an inscribed angle intercepts a
semicircle, the angle is a right angle.
If a right triangle is inscribed in a circle, then the
hypotenuse is a diameter of the circle. Conversely, if
one side of an inscribed triangle is a diameter of the
circle, then the triangle is a right triangle and the
angle opposite the diameter is the right angle.
B is a right angle if and only if AC is a diameter of
the circle.
A
B
C
quadrilateral can be
inscribed in a circle if
and only if its opposite
angles are
supplementary.
 D, E, F, and G lie on
some circle, C, if and
only if mD + mF =
180° and mE + mG =
180°
F
A
E
C
G
D
B
 Find
the value of
each variable.
 AB
is a diameter.
So, C is a right
angle and mC = 90°
 2x° = 90°
 x = 45
Q
A
•2x°
C
D
•z°
 Find
the value of
each variable.
is inscribed in
a circle, so opposite
angles are
supplementary.
 mD + mF = 180°
 z + 80 = 180
 z = 100
E
•120°
•80°
 DEFG
•y°
G
F
D
•z°
 Find
the value of
each variable.
is inscribed in
a circle, so opposite
angles are
supplementary.
 mE + mG = 180°
 y + 120 = 180
 y = 60
E
•120°
•80°
 DEFG
•y°
G
F