Back to Shadows…
A total solar eclipse occurs when the moon passes between the earth and the
sun, and the darkest shadow of the moon, called the umbra, hits the surface of
the earth. If the umbra does not hit the surface, as shown in the figure below,
then a total solar eclipse is not possible. In other words, for a total solar
eclipse to occur, point P must lie inside the circle that represents the earth.
Assume the diameter of the sun is 870,000 miles, the diameter of the moon is
2160 miles, the diameter of the earth is 7920 miles, and the distance from the
center of the sun to the center of the earth is approximately 93,000,000 miles.
The distance from the moon to the earth varies, but the maximum distance
from the center of the moon to the center of the earth is 252,700 miles, and is
called the lunar apogee. How far is P from the center of the earth during lunar
apogee? Can there be a total solar eclipse during lunar apogee?
The umbra
435
A
1.08
B
P
3.96
C
Equally Wet
The Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a line
segment then it is equidistant from the endpoints of the
segment.
The Perpendicular Bisector Theorem Converse
If a point is equidistant from the endpoints of a segment,
then it lies on the perpendicular bisector of the segment.
To locate the center of the circle
that goes through all three
vertices of a triangle we have
been finding the point where
the perpendicular bisectors of
two of the sides intersect.
D
When three or more lines go
through the same point, they
are said to be CONCURRENT.
Prove that the perpendicular
bisector of the third side goes
through point D as well, so that
all three perpendicular
bisectors are concurrent.
Given
Circles
DF is the  bisector of AB
B
DG is the  bisector of AC
F
Prove :
Point D lies on the  bisector of BC
D
A
G
Statements
C
Reasons
Given
Circles
DF is the  bisector of AB
B
DG is the  bisector of AC
F
Prove :
Point D lies on the  bisector of BC
D
A
G
C
Statements
Reasons
BD  DA
DA  DC
BD  DC
D lies on the  bisector of BC
 Bisector Thm.
 Bisector Thm.
Substitution
 Bisector Thm Converse
Circles
B
All three
perpendicular
bisectors of go
through point D and
are concurrent?
F
D
A
G
C
Point D is called
the circumcenter
of the inscribed
triangle.
A person is in a plane h units above sea level. If the
radius of the earth is r, write a function for the
distance to the horizon (d) in terms of h and r.
h
d
D(h, r )  an expression in terms of h and r
d 2  h(h  2r )
d 2  h2  2hr
D(h, r )  h 2  2hr
Use 3960 miles as the radius of
the earth, and determine the
distance to the horizon if you
are at 12,000 feet (There are
5280 feet in one mile)
D(2.27,3960)  (2.27) 2  2(2.27)(3960)
D(2.27,3960)  134.1 miles
On Patrol
On Patrol
The Angle Bisector Theorem
If a point lies on an angles bisector then it is equidistant
from the sides of the angle.
The Angle Bisector Theorem Converse
If a point is equidistant from the sides of an angle, then it
lies on the angles bisector.
Circles
Given
CE bisects BCA
AE bisects BAC
B
C
E
Prove :
Point E lies on the bisector of ABC
A
Statements
Reasons
Circles
Given
CE bisects BCA
B
AE bisects BAC
G
F
C
E
D
A
Statements
Prove :
Point E lies on the bisector of ABC
Reasons
 Bisector Thm.
EF  ED
 Bisector Thm.
ED  EG
Substitution
EF  EG
E lies on the  bisector of ABC  Bisector Thm. Converse
Circles
B
G
F
C
All three angle
bisectors go through
point E and are
concurrent.
E
D
A
Point E is called
the incenter of
the circumscribed
triangle.
Are the medians
of a triangle
concurrent?
Draw BO so that it goes through point P and
prove that O is the midpoint of segment AC.
B
Also…
PO  12 PD
BP  PD
M
Point P is called the
Centroid of a Triangle
N
PO  12 BP
P
BP  23 BO
ADCP is a parallelogram
A
Midsegment Theorem.
O
AD MP
C
DC PN
D
What do you know
about the diagonals
of a parallelogram?
Introduce D so that
DP = PB
 AO  OC so BO is a median (and they are concurrent through P)
Are the altitudes
of a triangle
concurrent?
ADBC and BACE are parallelograms
AC  DB
AC  BE
DB  BE
 B is the midpoint of DE
BG lies on the  bisector of DE
D
E
B
>
H
A
Point H is the
Circumcenter of
Triangle FDE and
what is called the
Orthocenter of
Triangle ABC.
>
G
F
C
Since the blue
segments are all
perpendicular
bisectors of Triangle
FDE, they all must be
concurrent.
Back to Shadows…
The centers of the earth and moon are as far apart as they
can be (252.7 thousand miles) during the lunar apogee.
93000  252.7  92747.3
The umbra
435
A
1.08
92747.3
B
435
1.08

92747.3  x
x
435x  1.08(92747.3  x)
x  230.84
P
C
PC  BC  BP
PC  252.7  230.84
PC  21.86
Summary of the Four Centers of a Triangle
Name of Center
Circumcenter
Orthocenter
Point of Intersection
Significance
Perpendicular Bisectors Center of
Circumscribed Circle
Altitudes
Forms Orthocentric Set
Together with Vertices
Incenter
Angle Bisectors
Centroid
Medians
Center of Inscribed
Circle
Center of Gravity
Cars often have pulley systems running off
of the motor to power systems like the airconditioning, alternator, or power steering.
Find the lengths of the belts required in the
following diagrams.
The radius of the small pulley is
2 inches, the large pulley is 7
inches and the distance
between them is 15 inches.
The radius of the small pulley
is 5 inches, the large pulley is
8 inches and the distance
between them is 16 inches.