Name Class Date 8.4 Midsegments of Triangles
Essential Question: H
ow are the segments that join the midpoints of a triangle’s sides related
to the triangle’s sides?
Resource
Locker
Explore Investigating Midsegments of a Triangle
The midsegment of a triangle is a line segment that connects the midpoints of two sides of
the triangle. Every triangle has three midsegments. Midsegments are often used_
to add
rigidity to structures. In the support for the garden swing shown, the crossbar DE​
​   
is a
midsegment of △ABC
You can use a compass and straightedge to construct the midsegments of a triangle.
A
Sketch a scalene triangle and label
the vertices A, B, and C.
A
B
Use a compass to find the midpoint
of AB​
​  ¯. Label the midpoint D.
A
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D
B
C
B
C
¯. Label
Use a compass to find the midpoint of ​ AC​
the midpoint E.
D
_ _
Use a straightedge to draw DE​
​  . DE​
​   
is one of the
midsegments of the triangle.
A
A
D
E
D
B
E
C
E
B
C
C
Repeat the process to _
find the other two midsegments of △ABC. You may want to
label the midpoint of BC​
​   
as F.
Module 8
395
Lesson 4
Reflect
1.
_
_
Use a ruler
the length of DE to the length of BC. What does this tell you
_
_ to compare
about DE and BC?
2.
_
Use a_
protractor to compare m∠ADE and m∠ABC. What does this tell you about DE
and BC? Explain.
3.
Compare your results with your class. Then state a conjecture about a midsegment of
a triangle.
Explain 1
Describing Midsegments on a
Coordinate Grid
You can confirm your conjecture about midsegments using the formulas for the midpoint,
slope, and distance.
Example 1
A
Show that the given midsegment of the triangle is parallel to the third side
of the triangle and is half as long as the third side.
The vertices of △_
GHI are G(-7, -1), H(-5, 5), and I(1, 3). J is
_
_
the
midpoint
of
GH
,
and
K
is
the
midpoint
of
IH
.
Show
that
JK ∥
_
_
GI and JK = __12 GI. Sketch JK.
(
K 4
)
y1 + y2
x1 + x2 _
,
, to find the
Step 1 Use the midpoint formula, _
2
2
coordinates of J and K.
)
(
I
J
x
-8
G
)
_
-5 + 1 5 + 3
The midpoint of IH is _, _ = (-2, 4). Graph
2
2
_
and label this point K. Use a straightedge to draw JK.
(
y
-4
-2
0
2
-2
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(
_
-1 + 5
-7 - 5 , _
= (-6, 2).
The midpoint of GH is _
2
2
Graph and label this point J.
6
H
-4
)
_
_
y2 - y1
Step 2 Use _
x 2 - x 1 to compare the slopes of JK and GI.
_ 3 - (-1)
_
1
1 Slope of GI
4-2
Slope of JK = _
=_
=_=_
2
(-7) 2
1
-2 - (-6)
_ _
Since the slopes are the same, JK ∥ GI.
____
_
_
Step 3 Use √(x 2 - x 1) 2 + (y 2 - y 1) 2 to compare the lengths of JK and GI.
____
JK = √( -2 - (-6) ) 2 + (4 - 2) 2 =
GI =
_
√ 20
_
= 2√5
―――――――――― = √_
_
80 = 4√5
√(1 - (-7)) + (3 - (-1))
2
2
_
1 (4√_
1 GI.
Since 2√5 = _
5 ), JK = _
2
2
Module 8
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Lesson 4
B
The vertices of △LMN are L​(2, 7)​, M​(10, 9)​, and N​(8, 1)​. P is the
_
_
midpoint of LM​
​  
, and Q is the midpoint of MN​
​  
.
_ _
_
Show that PQ​
​  ∥ LN​
​   
and PQ = ​  1  ​LN. Sketch PQ​
.
​  
2
_ 
(
_ _
7+
2+
 ​, ​  _
 ​ 
 = ​
Step 1 The midpoint of LM​
​  = ​    
2
2
Graph and label this point P.
(
,
)
10
(
)
L
6
)
​. 4
2
0
2
4
x
10
_ __
Slope of LN​
​  =   
​ 
 ​ =
-
_
_
Since the slopes are the same, PQ​
​   
and LN​
​  are .
____
√(
)
2
__
√
- 6 ​ ​ ​ + (​​  5 - 8 )​ ​ ​ ​ = ​  
Step 3 PQ = ​ ​​  
2
_____
√( 
) ( 
2
_
_
+ 9 ​ 
=√
​ 18 ​ = 3​√2 ​ 
___
) √
2
√
_
+
  ​ ​ ​ + ​​
  ​ ​ ​ ​ = ​   
 ​ = ​ LN = ​ ​​   
Since
N
8
6
_
​. Graph and label this point Q. Use a straightedge to draw PQ​
​  .
,
_ _
Step 2 Slope of PQ​
​   
= ​  5 - 8  
 ​ 
=
9-
M
8
_
+
+
 ​
The midpoint of NM​
​  
= ​​  __
  
 ​, 
​  __
 
 
​
2
2
= ​
y
_
√​
(  √​  ​ )​,
1
_
 ​ = ​  2 ​​ 
_
 
=_
​  1 ​ 
2
_
√
= 6 ​
 ​ 
 ​  
.
_
_
​  
.
The length of PQ​
​  is the length of LN​
Your Turn
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4.
The vertices _
of △XYZ are X​(3, 7)​, Y​(9, 11)​,_
and Z​(7, 1)​. U is the
midpoint
of
​
XY​
,
 
and
W
is
the
midpoint
of
XZ​
​
 
. Show
_ _
_ that
1
_
∥
​ 
UW​
  YZ​
​   
and UW = ​   ​ YZ. Sketch △XYZ and UW​
​  
.
2
y
10
8
6
4
2
0
Module 8
397
x
Z
2
4
6
8
10
Lesson 4
Explain 2
Using the Triangle Midsegment
Theorem
The relationship you have been exploring is true for the three midsegments of every triangle.
Triangle Midsegment Theorem
The segment joining the midpoints of two sides of a triangle is parallel to the third
side, and its length is half the length of that side.
You explored this theorem in Example 1 and will be proving it later in this course.
Example 2
A
Use triangle RST.
Find UW.
S
By the Triangle
Midsegment Theorem,_the length of
_
midsegment UW is half the length of ST.
1 ST
UW = _
2
1 (10.2)
UW = _
2
UW = 5.1
U
41°
5.6
V
10.2
R
W
B
T
Complete the reasoning to find m ∠SVU.
_ _
UWǁST
m ∠SVU = m ∠VUW
Substitute
for
m ∠SVU =
Reflect
How do you know to which side of a triangle a midsegment is parallel?
Your Turn
6.
P
J
Find JL, PM, and m ∠MLK.
K
39
105° M
N
95
L
Elaborate
7.
¯ is NOT a midsegment of the triangle.
Discussion Explain why XY
6
5
Module 8
398
X
6
Y
5
Lesson 4
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5.
8.
Essential Question Check–In Explain how the perimeter of △DEF
compares to that of △ABC.
A
F
C
D
E
B
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Evaluate: Homework and Practice
1.
Use a compass and a ruler or geometry software to construct an obtuse triangle.
Label the vertices. Choose two sides and construct the midpoint of each side; then
label and draw the midsegment. Describe the relationship between the length of the
midsegment and the length of the third side.
2.
)
The vertices of △WXY are W(-4, 1), X( 0, -5
Y 4, -1). A is the midpoint of
_, and
_
_
_(
1 WX.
WY, and B is the midpoint of XY. Show that AB ǁWX and AB = _
2
3.
), and H(-5, –2). X is the midpoint of
The
_ vertices of △FGH are F(-1,
_
_ 1), G( -5, 4_
FG, and Y is the midpoint of FH. Show that XY ǁGH and XY = __12 GH.
4.
• Online Homework
• Hints and Help
• Extra Practice
_
One of the vertices of △PQR is P(3, –2). The midpoint of PQ is M(4, 0).
_
_ _
1 PR.
The midpoint of QR is N(7, 1). Show that MN ǁ PR and MN = _
2
Module 8
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Lesson 4
5.
(  )
_
One
of
the
vertices
of
△
ABC
is
A​
0,
0
.
​
The
midpoint
of
AC​
​ 
 
is J ​ _
​  3 ​ ,  2  ​. The midpoint
(
)
_
_ _
2
1
of BC​
​   
is K (​ 4, 2)​. Show that JK​
​   ǁ​ BA​  
and JK = _
​   ​  BA.
2
Find each measure.
A
X
B
68°
Y
4.6
C
Z
15.8
6.
XY
7.
BZ
8.
AX
9.
m∠YZC
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10. m∠BXY
Module 8
400
Lesson 4
Algebra Find the value of n in each triangle.
11.
12.
6n
11.3
48
n + 4.2
13.
14.
14n
n + 12
4n + 9
6n
15. Line segment XY is a midsegment of △MNP. Determine whether each of the
following statements is true or false. Select the correct answer for each lettered part.
N
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X
M
Y
P
a. MP = 2XY
1  ​X
b. MP = ​ _
  Y
2
c. MX = XN
d. MX = _
​  1 ​ NX
2
e. NX = YN
1  ​M
  P
f. XY = ​ _
2
True
False
True
False
True
False
True
False
True
False
True
False
16. What do you know about two of the midsegments in an isosceles triangle? Explain.
Module 8
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Lesson 4
17. Suppose you know that the midsegments of a triangle are all 2 units long. What kind
of triangle is it?
_ _
_
​  
, BC​
​  
, and AC​
​  
18. In △ABC, m∠A = 80°, m∠B = 60°, m∠C = 40°. The midpoints of AB​
are D, E, and F, respectively. Which midsegment will be the longest? Explain how you
know.
19. Draw Conclusions Carl’s Construction is building a pavilion
with an A-frame roof at the local park. Carl has constructed two
triangular frames for the front and back of the roof, similar to
△
_ABC in the diagram. The base of each frame, represented by ​
AC​, is 36 feet _
long. He_
needs to insert a crossbar connecting the
midpoints of AB​
​  and BC​
​  
, for each frame. He has 32 feet of timber
left after constructing the front and back triangles. Is this enough
to construct the crossbar for both the front and back frame?
Explain.
B
D
E
A
C
2​
(QR)​= AB
(25)​ = AB
2​
50 = AB
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Polianskyi
Igor/Shutterstock
AB is a midsegment in △PQR.
20. Critique Reasoning Line segment
_
Kayla calculated the length of AB​
​  
. Her work is shown below. Is her
answer correct? If not, explain her error.
R
B
25
Q
A
P
21. Using words or diagrams, tell how to construct a midsegment using only a
straightedge and a compass.
Module 8
402
Lesson 4
H.O.T. Focus on Higher Order Thinking
22. Multi–Step A city park will be shaped like a right triangle,
_
and _
there will be two pathways for pedestrians, shown by VT​
​  
and ​VW​ 
in the diagram. The park planner only wrote two
lengths on his sketch as shown. Based on the diagram, what
will be the lengths of the two pathways?
Y
30 yd
X
V
T
24 yd
W
Z
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Carey
Hope/iStockPhoto.com
23. Communicate Mathematical Ideas △XYZ is the midsegment of △PQR.
Write a congruence statement involving all four of the smaller triangles. What
is the relationship between the area of △XYZ and △PQR?
Q
Y
X
P
R
Z
_
_
​  is a
24. Copy the diagram shown. AB​
​  is a midsegment of △XYZ. CD​
midsegment of △ABZ.
Z
AB​? 
What is the ratio of AB to XY?
a. What is the length of​ ¯
C
D
A
B
CD​? 
What is the ratio of CD to XY?
b. What is the length of​ ¯
_
​ 34 ​the distance from point Z
c. Draw ​EF​ such that points E and F are __
to _
points X and Y. What is the ratio of EF to XY? What is the length
of ​EF​ 
?
X
64
Y
d. Make a conjecture about the length of non-midsegments when compared to the
length of the third side.
Module 8
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Lesson 4
Lesson Performance Task
The figure shows part of a common roof design
using
stable triangular trusses.
_ and_
_ _
_very
_strong
Points B, C, D, F, G, and I are midpoints of AC​
​ , AE​
​  
, CE​
​  
, GE​
​  
, HE​
​  and AH​
​  respectively. What is
the total length of all the stabilizing bars inside △AEH? Explain how you found the answer.
A
J
20 ft
B
C
I
4 ft
D
H
G
F
14 ft
E
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Module 8
404
Lesson 4