Transcript 5.1 Midsegment Theorem 5.2 Use Perpendicular Bisectors

5.1 Midsegment Theorem and Coordinate Proof
Objectives:
1. To discover and use the Midsegment
Theorem
2. To write a coordinate proof
Midsegment
A midsegment of a
triangle is a
segment that
connects the
midpoints of two
sides of the triangle.
Every triangle has 3
midsegments.
Midsegment
A midsegment of a
triangle is a
segment that
connects the
midpoints of two
sides of the triangle.
Every triangle has 3
midsegments.
Example 1
Graph ΔACE with
coordinates
A(-1, -1), C(3, 5),
and E(7, -5).
Graph the
midsegment MS
that connects the
midpoints of AC
and CE.
6
C
4
2
M
S
5
A
-2
-4
E
Example 1
Now find the slope
and length of MS
and AE. What do
you notice about
the midsegment
and the third side
of the triangle?
6
C
4
2
M
S
5
A
-2
-4
E
Midsegment Theorem
The segment
connecting the
midpoints of two
sides of a triangle
is parallel to the
third side and is
half as long as
that side.
Example 2
The diagram shows an illustration of a roof
truss, where UV and VW are midsegments
of ΔRST. Find UV and RS.
Example 3
1.
2.
Coordinate Proof
Coordinate proofs are easy. You just have
to conveniently place your geometric
figure in the coordinate plane and use
variables to represent each vertex.
– These variables, of course, can represent any
and all cases.
– When the shape is in the coordinate plane,
it’s just a simple matter of using formulas for
distance, slope, midpoints, etc.
Example 4
Place a rectangle in
the coordinate
plane in such a way
that it is convenient
for finding side
lengths. Assign
variables for the
coordinates of each
vertex.
Example 4
Convenient
placement usually
involves using the
origin as a vertex
and lining up one
or more sides of
the shape on the
x- or y-axis.
Example 5
Place a triangle in the
coordinate plane in
such a way that it is
convenient for
finding side
lengths. Assign
variables for the
coordinates of each
vertex.
Example 6
Place the figure in the coordinate plane in a
convenient way. Assign coordinates to
each vertex.
1. Right triangle: leg lengths are 5 units and
3 units
2. Isosceles Right triangle: leg length is 10
units
Example 7
A square has vertices
(0, 0), (m, 0), and
(0, m). Find the
fourth vertex.
y
0, m 
m, m 
x
0, 0
m, 0
Example 8
Find the missing
coordinates. The
show that the
statement is true.
Example 9
Write a coordinate proof for the Midsegment
Theorem.
y
Given: MS is a midsegment of
ΔOWL
W b, c 
M
S
Prove: MS || OL and MS = ½OL
x
O 0, 0
L a, 0
Example 10
Explain why the choice of variables below
might be slightly more convenient.
y
Given: MS is a midsegment of
ΔOWL
W 2b, 2c 
M
S
Prove: MS || OL and MS = ½OL
x
O 0, 0
L 2a, 0
Perpendicular Bisector
A segment, ray, line,
or plane that is
perpendicular to a
segment at its
midpoint is called a
perpendicular
bisector.
Equidistant
A point is equidistant
from two figures if
the point is the
same distance from
each figure.
Examples: midpoints
and parallel lines
5.2: Special Segments
Objectives:
1. To use and define perpendicular
bisectors, angle bisectors,
2. To discover, use, and prove various
theorems about perpendicular bisectors
and angle bisectors
Perpendicular Bisector Theorem
In a plane, if a point is
on the perpendicular
bisector of a
segment, then it is
equidistant from the
endpoints of the
segment.
Converse of Perpendicular Bisector Theorem
In a plane, if a point is
equidistant from the
endpoints of a
segment, then it is
on the perpendicular
bisector of the
segment.
Example 1
Plan a proof for the Perpendicular Bisector
Theorem.
Example 2
BD is the perpendicular bisector of AC. Find
AD.
Example 3
Find the values of x and y.
Angle Bisector
An angle bisector is a
ray that divides an
angle into two
congruent angles.
Angle Bisector Theorem
If a point is on the bisector of an angle, then
it is equidistant from the two sides of the
angle.
Example 4
A soccer goalie’s position relative to the ball
and goalposts forms congruent angles, as
shown. Will the goalie have to move
farther to block a shot toward the right goal
post or the left one?
Example 5
Find the value of x.
Example 6
Find the measure of <GFJ.
It’s not the Angle Bisector Theorem that could help
us answer this question. It’s the converse. If it’s true.
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is
equidistant from the sides of the angle,
then it lies on the bisector of the angle.
Example 7
For what value of x does P lie on the bisector
of <A?
Warm-Up
Three or more lines
that intersect at the
same point are
called concurrent
lines. The point of
intersection is called
the point of
concurrency.
E
C
B
G
A
D
F
Example 1
Are the lines represented by the equations
below concurrent? If so, find the point of
concurrency.
x+y=7
x + 2y = 10
x-y=1
Points of Concurrency
Objectives:
1. To define various points of concurrency
2. To discover, use, and prove various
theorems about points of concurrency
Concurrency of Medians Theorem
The medians of a
triangle intersect at
a point that is twothirds of the
distance from each
vertex to the
midpoint of the
opposite side.
Centroid
The three medians of a
triangle are
concurrent. The point
of concurrency is an
interior point called
the centroid. It is the
balancing point or
center of gravity of the
triangle.
Example 2
In ΔRST, Q is the centroid and SQ = 8. Find
QW and SW.
Circumcenter
Concurrency of
Perpendicular Bisectors
of a Triangle Theorem
The perpendicular
bisectors of a triangle
intersect at a point that
is equidistant from the
vertices of the triangle.
Circumcenter
The point of concurrency of the three
perpendicular bisectors of a triangle is
called the circumcenter of the triangle.
In each diagram, the circle circumscribes the triangle.
Incenter
Concurrency of Angle
Bisectors of a
Triangle Theorem
The angle bisectors of
a triangle intersect
at a point that is
equidistant from the
sides of the triangle.
Incenter
The point of concurrency of the three angle
bisectors of a triangle is called the
incenter of the triangle.
In the diagram, the circle is inscribed within the triangle.
Orthocenter
Concurrency of
Altitudes of a
Triangle Theorem
The lines containing
the altitudes of a
triangle are
concurrent.
G
Orthocenter
The point of concurrency of all three altitudes
of a triangle is called the orthocenter of
the triangle.
The orthocenter, P, can be inside, on, or outside
of a triangle depending on whether it is acute,
right, or obtuse, respectively.