Transcript 5.4 Midsegment Theorem Geometry Mrs. Spitz Fall 2004 Objectives:   Identify the midsegments of a triangle. Use properties of midsegments of a triangle.

5.4 Midsegment Theorem
Geometry
Mrs. Spitz
Fall 2004
Objectives:


Identify the midsegments of a triangle.
Use properties of midsegments of a
triangle.
Assignment

Pgs. 290-291 #1-18, 21-22, 26-29
Using Midsegments of a
Triangle

In lessons 5.2 and 5.3, you studied four
special types of segments of a triangle:





Perpendicular bisectors
Angle bisectors
Medians and
Altitudes
A midsegment of a triangle is a segment that
connects the midpoints of two sides of a
triangle.
How?

1.
2.
3.
4.
You can form the three midsegments of a triangle
by tracing the triangle on paper, cutting it out, and
folding it as shown.
Fold one vertex onto another to find one midpoint.
Repeat the process to find the other two midpoints.
Fold a segment that contains two of the midpoints.
Fold the remaining two midsegments of the
triangle.
Ex. 1: Using midsegments
K (4, 5)


Show that the
midsegment MN is
parallel to side JK
and is half as long.
Use the midpoint
formula.
4
J (-2, 3)
2
N
M
5
L (6, -1)
-2
-4
Solution:
M= -2+6 , 3+(-1)
2
2
M = (2, 1)
And
N = 4+6 , 5+(-1)
2
2
N = (5, 2)
Next find the slopes of JK and MN.
Slope of JK = 5 – 3 = 2 = 1
4-(-2) 6
3
Slope of MN= 2 – 1 = 1
5–2 3
►Because their slopes are equal,
JK and MN are parallel. You
can use the Distance Formula
to show that MN = √10 and JK =
√40 = 2√10. So MN is half as
long as JK.
Theorem 5.9: Midsegment
Theorem



The segment
connecting the
midpoints of two
sides of a triangle is
parallel to the third
side and is half as
long.
DE ║ AB, and
DE = ½ AB
C
D
A
E
B
Ex. 2: Using the Midsegment
Theorem





UW and VW are midsegments of
∆RST. Find UW and RT.
SOLUTION:
UW = ½(RS) = ½ (12) = 6
RT = 2(VW) = 2(8) = 16
A coordinate proof of Theorem 5.9 for
one midsegment of a triangle is given on
the next slide. Exercises 23-25 ask for
proofs about the other two midsegments.
To set up a coordinate proof, remember
to place the figure in a convenient
location.
R
U
12
V
8
T
W
S
Ex. 3: Proving Theorem 5.9
4

Write a coordinate proof of the
Midsegment Theorem.
3
C (2a, 2b)
2
1
Place points A, B, and C in
convenient locations in a
coordinate plane, as shown.
Use the Midpoint formula to
find the coordinate of
midpoints D and E.
D
A (0, 0)
-1
E
2B ( 2c, 0)
4
6
Ex. 3: Proving Theorem 5.9
D = 2a + 0 , 2b + 0 = a, b
2
2
E = 2a + 2c , 2b + 0 = a+c, b
2
2
Find the slope of midsegment DE. Points D and E have the
same y-coordinates, so the slope of DE is 0.
►AB also has a slope of 0, so the slopes are equal and DE
and AB are parallel.
Now what?
Calculate the lengths of DE and AB. The
segments are both horizontal, so their lengths
are given by the absolute values of the
differences of their x-coordinates.
AB = |2c – 0| = 2c
DE = |a + c – a | = c
►The length of DE is half the length of AB.
Objective 2: Using properties
of Midsegments

Suppose you are given only the three
midpoints of the sides of a triangle. Is
it possible to draw the original triangle?
Example 4 shows one method.
8
What?
6
PLOT the midpoints
on the coordinate
plane.
CONNECT these
midpoints to form
the midsegments
LN, MN, and ML.
FIND the slopes of
the midsegments.
Use the slope
formula as shown.
A (3, 5)
4
Slope MN = 0.33
N
Slope = 1/3
Slope NL = 2.00
M
B (7, 3)
2
Slope = 1/2
Slope LM = -0.50
L
C (1, 1)
5
8
What?
6
ML = 3-2 = -1
2-4
2
MN = 4-3 = 1
5-2
A (3, 5)
3
4
N
Slope = 1/3
LN = 4-2 = 2 = 2
5-4
Slope MN = 0.33
Slope NL = 2.00
M
B (7, 3)
1
Each midsegment contains
two of the unknown
triangle’s midpoints and is
parallel to the side that
contains the third midpoint.
So you know a point on each
side of the triangle and the
slope of each side.
2
Slope = 1/2
Slope LM = -0.50
L
C (1, 1)
5
8
What?
6
►DRAW the lines that
contain the three sides.
The lines intersect at 3
different points.
A (3, 5)
4
Slope NL = 2.00
M
B (7, 3)
B (7, 3)
The perimeter formed by the
three midsegments of a
triangle is half the perimeter
of the original triangle shown
in example #5.
N
Slope = 1/3
A (3, 5)
C (1, 1)
Slope MN = 0.33
2
Slope = 1/2
Slope LM = -0.50
L
C (1, 1)
5
Ex. 5: Perimeter of
Midsegment Triangle
DF = ½ AB = ½ (10) = 5
EF = ½ AC = ½ (10) = 5
ED = ½ BC=½ (14.2)= 7.1
►The perimeter of ∆DEF is 5 + 5
+ 7.1, or 17.1. The perimeter
of ∆ABC is 10 + 10 + 14.2, or
34.2, so the perimeter of the
triangle formed by the
midsegments is half the
perimeter of the original
triangle.
10 cm
A
B
E
10 cm
D
C
14.2 cm
F