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5.1 Midsegment Theorem and Coordinate Proof Objectives: Assignment: P. 298-301: 1-6, 8, 10,

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5.1 Midsegment Theorem and Coordinate Proof Objectives: Assignment: P. 298-301: 1-6, 8, 10,
5.1 Midsegment Theorem and Coordinate Proof
Objectives:
1. To discover, use,
and prove the
Midsegment
Theorem
2. To write a coordinate
proof
Assignment:
• P. 298-301: 1-6, 8, 10,
12-19 some, 20, 21,
24, 29, 30, 36, 37, 47,
48
• Challenge Problems
Objective 1
You will be able to discover,
use, and prove the
Midsegment Theorem
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
Midsegment
A midsegment of
a triangle is a
segment that
connects the
midpoints of two
sides of the
triangle.
Every triangle has 3 midsegments
Investigation 1
1. On a piece of patty
paper, draw a
large acute ΔABC.
2. Find the midpoints
of each side by
putting two vertices
on top of each
other and pinching
the midpoint.
Investigation 1
3. Label the
midpoints M, N,
and P. Draw the
three midsegments
of your triangle by
connecting the
midpoints of each
side.
Investigation 1
4. Use another piece
of patty paper to
trace off ΔAMP.
M
A
P
Investigation 1
5. Compare all the
small triangles.
What do you notice
about the length of
a midsegment and
the opposite side of
the triangle? What
kind of lines do
they appear to be?
M
A
P
Investigation 1
5. Compare all the
small triangles.
What do you notice
about the length of
a midsegment and
the opposite side of
the triangle? What
kind of lines do
they appear to be?
M
A
P
Investigation 1
A
P
M
5. Compare all the
small triangles.
What do you notice
about the length of
a midsegment and
the opposite side of
the triangle? What
kind of lines do
they appear to be?
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  and  are midsegments
of ΔRST. Find UV and RS.
Example 3
1.
2.
Deep, Penetrating Questions
How many examples did we look at to come
up with our Theorem?
Is that enough?
How could we prove this theorem?
Where could we prove this theorem?
Objective 2
You will be able to write a
coordinate proof
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 to
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
5.1 Midsegment Theorem and Coordinate Proof
Objectives:
1. To discover, use,
and prove the
Midsegment
Theorem
2. To write a coordinate
proof
Assignment:
• P. 298-301: 1-6, 8, 10,
12-19 some, 20, 21,
24, 29, 30, 36, 37, 47,
48
• Challenge Problems
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