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Honors Geometry
Lesson 5.1
Midsegment Theorem
What You Should Learn
Why You Should Learn It
• Goal 1: How to identify and construct
the midsegments of a triangle
• Goal 2: How to use properties of
midsegments to solve real-life
problems
Midsegment
• The midsegment of a triangle is a
segment that connects the midpoints
of two sides of the triangle
Lesson Investigation
Observations about midsegments
Midsegment is
half the length
of the side it is
parallel to
Midsegment is
parallel to the third
side of the triangle
Midsegment Theorem
• The segment connecting the
midpoints of two sides of a triangle
is parallel to the third side and is
half its length
Example 1
Illustrating the Midsegment Theorem
• Show that the
midsegment MN
is parallel to the
side JK and half
its length
Example 1 Solution
Illustrating the Midsegment Theorem
• Begin by using the Midpoint Formula
-2+6 3  1
M=(
,
)  (2,1)
2
2
4+6 5  1
N=(
,
)  (5, 2)
2
2
Example 1 Solution
Illustrating the Midsegment Theorem
• Now find the slopes of JK and MN
5-3
2 1
slope JK =
 
4-(-2) 6 3
2-1 1
slope MN =

5-2 3
Because JK and MN have
the same slope, they must be
parallel
(5,2)
(2,1)
Example 1 Solution
Illustrating the Midsegment Theorem
• Now use the distance formula to find the
lengths JK and MN
distance JK = (4-(-2)) 2  (5  3) 2
36+4  40  4  10
2 10
distance MN = (5-2) 2  (2  1) 2
(5,2)
9+1
10
(2,1)
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: 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
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