Download 5-4 Reteach The Triangle Midsegment Theorem

Name
LESSON
5-4
Date
Class
Reteach
The Triangle Midsegment Theorem
A midsegment of a triangle joins the midpoints of two sides of the triangle.
Every triangle has three midsegments.
$
_
_
RS is a midsegment
of CDE.
R is the midpoint of _
CD .
S is the midpoint of CE .
#
3
%
_
Use the figure for Exercises 1–4. AB is a midsegment of RST.
_
1. What is_
the slope of midsegment AB and the slope
of side ST ?
Y
3 1; 1
!
_
_
_ _
2. What can you conclude about AB and ST ?
Since the slopes are the same, AB ST.
X
4 " 2
3. Find AB and ST.
AB 22 , ST 4 2
_
_
4. Compare the lengths of AB and ST .
1 ST or ST 2AB
AB __
2
Use MNP for Exercises 5–7.
_
5. UV is a midsegment of MNP. Find the
coordinates of U and V.
Y
-
. U (1, 3), V (3, 2)
_
_
6. Show that UV MN .
_
1 and the slope of
The slope of UV __
4
_
1. Since the slopes are the
MN __
_4 _
same, UV MN.
5
6
0 X
1 MN.
7. Show that UV __
2
1 (2
1 MN.
UV 17 and MN 2 17 . Since 17 __
17 ), UV __
2
2
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
30
Holt Geometry
Name
LESSON
5-4
Date
Class
Reteach
The Triangle Midsegment Theorem
continued
Theorem
Example
0
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to
a side of the triangle, and its length is
half the length of that side.
1
,
.
_
Given: PQ is a midsegment of LMN.
_ _
1 LN
Conclusion: PQ LN , PQ __
2
"
You can use the Triangle Midsegment Theorem to
find various measures in ABC.
1 AC
HJ __
Midsegment Thm.
2
1 (12)
HJ __
Substitute 12 for AC.
2
HJ 6
Simplify.
1 AB
JK __
2
__
4 1 AB
2
8 AB
_
—
(
*
!
+
#
_
Midsegment Thm.
HJ || AC
Midsegment Thm.
Substitute 4 for JK.
mBCA mBJH
Corr. Thm.
Simplify.
mBCA 35°
Substitute 35° for mBJH.
Find each measure.
8. VX 23
9. HJ 54
(
—
8
7
92°
10. mVXJ 11. XJ 13. DE *
6
27
Find each measure.
12. ST '
3
%
72
$
22
14. mDES 48°
15. mRCD 48°
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
4
—
#
2
31
Holt Geometry
Name
LESSON
5-4
Date
Class
Name
Practice A
LESSON
5-4
The Triangle Midsegment Theorem
Use the Triangle Midsegment Theorem
to name parts of the figure for
Exercises 1–5.
_
�
�
_
3. a segment that has the same length as BD
_
4. a segment that has half the length of AC
_
�
� ������ �
�
6. Use the Midpoint Formula to find the coordinates of G.
(
7. Use the Midpoint Formula to find the coordinates of H.
(
_
0
3
,
10. If two segments have
same slope, then the segments
_ the_
are parallel. Are DF and GH parallel?
�
���
12
16. PU
12
18. m�SUR
Name
LESSON
5-4
22
19. m�PRQ
55°
1522.5 mi
Date
_
Holt Geometry
Class
Reasons
1. Given
2. Midsegment Theorem
3. Definition of perimeter
4. Substitution
28
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
5-4
Date
Class
Holt Geometry
Reteach
The Triangle Midsegment Theorem
A midsegment of a triangle joins the midpoints of two sides of the triangle.
Every triangle has three midsegments.
�
� (0, 2�)
�
�
�
�
�
�(0, 0)
2ab
�
�
4. The perimeter of �STU � _1_ PQ � _1_ QR
2
2
� _1_ RP.
_
_
� (2�, 0)
RS is a midsegment
of �CDE.
R is the midpoint of _
CD.
S is the midpoint of CE.
D (0, b), E (a, b), F (a, 0)
�
3. Pedro knows it will be easy to find the area of �EFD if �DEF is a right angle. Write a
proof that �DEF � �A.
_
4.
�
3. The perimeter of �STU � ST � TU � US.
LESSON
Pedro has a hunch about the area of midsegment triangles.
He is a careful student, so he investigates in a methodical
manner. First Pedro draws a right triangle because he
knows it will be easy to calculate the area.
�
�
2
The Triangle Midsegment Theorem
2. Find the coordinates of the midpoints D, E, and F.
�
5. The perimeter �STU � _1_ (PQ � QR � RP) 5. Distributive Property
2
of �
Practice C
1. Find the area of �ABC.
3045 mi
Possible answer:
�
55°
27
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
San Juan
8. Find the perimeter of the midsegment triangle within
the Bermuda Triangle.
_ _
�
17. m�SUP
Bermuda
Miami
7. Use the distances in the chart to find the perimeter of
the Bermuda Triangle.
Statements
_ _
_
1. US, ST, and TU are midsegments of
�PQR.
2. ST � _1_ PQ, TU � _1_ QR, US � _1_ RP
2
2
2
15. QR
125°
�
Dist.
(mi)
Miami to San Juan
1038
Miami to Bermuda
1042
Bermuda to San Juan � 965
10. Given: US , ST , and TU are midsegments of �PQR.
Prove: The perimeter of �STU � _1_(PQ � QR � RP).
2
�
14. ST
�
�
It is half the perimeter of the Bermuda Triangle.
��
��
�
58°
6. m�D
9. How does the perimeter of the midsegment triangle compare to
the perimeter of the Bermuda Triangle?
�
�
58°
58°
4. m�HIF
122°
18.2
�
17.5
�
Write a two-column proof that the perimeter of a
midsegment triangle is half the perimeter of the triangle.
yes
6
3
yes
11. Use the Distance Formula to find DF.
Use the Triangle Midsegment Theorem
and the figure for Exercises 14–19.
Find each measure.
)
)
0
0
_
9. Use the Slope Formula to find the slope of GH.
12. Use the Distance Formula to find GH.
13. Does GH � _1_ DF ?
2
2
2
,
8. Use the Slope Formula to find the slope of DF.
9.1
�
35
2. DF
The Bermuda Triangle is a region in the
Atlantic Ocean off the southeast coast of
the United States. The triangle is bounded
by Miami, Florida; San Juan, Puerto Rico;
and Bermuda. In the figure, the dotted lines
are midsegments.
�
�������
�
9.1
5. m�HGD
� ������
�
The Triangle Midsegment Theorem
3. GE
BC
5. a segment that has twice the length of EC
�
Practice B
1. HI
_
DE
_
DE
_
AD
_
DE
_
�
�
Complete Exercises
_6–13 to show
that
_ midsegment GH is parallel to
DF and that GH � _1_ DF.
2
Class
Use the figure for Exercises 1–6. Find each measure.
�
1. a midsegment of �ABC
2. a segment parallel to AC
Date
�
�
_
Possible answer: F is the midpoint of AC, so AF � _1_ AC. DE � _1_ AC
2
_ _ _ _2
by the Midsegment Theorem, so AF � DE. DE � AF by the Midsegment
Theorem and
_and �AFD are alternate interior angles, so �EDF
_�EDF
� �AFD. DF � DF by the Reflexive Property, thus �EFD � �ADF.
By CPCTC, �DEF � �A.
_1_ ab
2
Find the area of �EFD.
Use the figure for Exercises 1–4. AB is a midsegment of �RST.
_
1. What is_
the slope of midsegment AB and the slope
of side ST ?
�
� ������
�
�1; �1
_
�������
_
_ _
�
2. What can you conclude about AB and ST ?
Since the slopes are the same, AB � ST.
�
�
� �������
��
� �������
��������
5. Compare the areas of �ABC and �EFD.
3. Find AB and ST.
�
�ABC has four times the area of �EFD.
�
AB � 2�2 , ST � 4�2
_1_ ab
6. Pedro has already shown that �EFD � �ADF.
Calculate the area of �ADF.
_
_
4. Compare the lengths of AB and ST .
2
AB � _1_ ST or ST � 2AB
7. Write a conjecture about congruent triangles and area.
2
Possible answer: Congruent triangles have equal area.
Use �MNP for Exercises 5–7.
_
Pedro already knows some things about the area of the
midsegment triangle of a right triangle. But he thinks he
can expand his theorem. Before he can get to that, however,
he has to show another property of triangles and area.
5. UV is a midsegment of �MNP. Find the
coordinates of U and V.
�
U (�1, 3), V (3, 2)
�
_
� ������
�
_
_
6. Show that UV � MN .
8. Find the area of �WXY, �WXZ, and �YXZ.
�
16; 6; 10
�
��������
�
�
�
The slope of UV � � _1_ and the slope of
4
_
MN � � _1_. Since the slopes are the
_4 _
same, UV � MN.
�
9. Compare the total of the areas of �WXZ and �YXZ to the area of �WXY.
Possible answer: The total of the areas of �WXZ and �YXZ is equal to
the area of �WXY.
7. Show that UV � _1_ MN.
� 2
10. Write a conjecture about the areas of triangles within a larger triangle.
�
�
��
�
�
� �������
�
UV � � 17 and MN � 2�17 . Since �17 � _1_ (2�17 ), UV � _1_ MN.
2
2
Possible answer: The area of a larger triangle is the sum of areas of the
�
�
�
triangles within it.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
29
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
73
30
Holt Geometry
Holt Geometry
Name
LESSON
5-4
Date
Class
Name
Reteach
LESSON
The Triangle Midsegment Theorem
5-4
continued
Theorem
�
�
_
Midsegment Thm.
Substitute 4 for JK.
m�BCA � m�BJH
Corr. � Thm.
Simplify.
m�BCA � 35°
Substitute 35° for m�BJH.
9. HJ �
54
case 1: EF � FG � 7 and EG � 8; case 2: EF � FG � 8 and EG � 6
��
92°
�
Find each measure.
12. ST �
72
13. DE �
22
�
�
�
���
��
14. m�DES �
48°
15. m�RCD �
48°
Two cases; the midsegment joins
If the midsegment joins the sides
the sides with lengths 12 and 18.
with lengths 12 and 18, then the
The midsegment joins the side
third side is 18. If the midsegment
with lengths 12 and x.
joins the side with lengths 12 and
x, then it is impossible to find the
�
length of the third side.
��
�
31
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
Date
Class
Holt Geometry
Problem Solving
LESSON
1. The vertices of
�JKL are J(�9, 2), K(10, 1),
_
parallel
and_L (5, 6). CD is the midsegment
_
to JK. What is the length of CD ? Round to
the nearest tenth.
5-4
2. In �QRS, QR � 2x � 5, RS � 3x � 1,
and SQ � 5x. What is the perimeter of
the midsegment triangle of �QRS ?
9.5
�
9.2 mi
�
_
3. Which midsegment is parallel to side QS ?
_
MN
4 mi
�
1.5 mi �
�
_
5. LN �
F 150�
G 140�
_
�
MR
�
�
�
H 110�
J 30�
�
�
_
8. The measure of �AYW is 50�. What is the
measure of �VWB ?
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
_
QM
6 cm
6. Draw the midsegments in �ABC.
6. In triangle HJK, m�H � 110�, m�J � 30°,
and m�K � 40�. If R is the midpoint
of
_
_
JK, and S is the midpoint of HK, what is
m�JRS ?
the midpoint of AB,
On the balance beam, V is
_
and W is the midpoint of YB.
_
7. The length of VW is 1 _7_ feet. What is AY ?
8
A _7_ ft
C 3 _3_ ft
8
4
15 ft
D 7 _1_ ft
B ___
16
2
F 45�
G 50�
_
_
4. If side RS is 12 cm, how long is LM?
1.7 mi
Choose the best answer.
Use the diagram for Exercises 7 and 8.
�
�
The Triangle Midsegment Theorem: A midsegment of a
triangle is parallel to a side of the triangle, and its length is
half the length of that side.
�(12, �3)
�
C 240 cm2
D 480 cm2
�
�LMN
�
8
� (1, �1)
4. The diagram at right shows horseback
riding trails. Point B
_
is the halfway point
_along path AC. Point
_D is the
_halfway
point along path CE. The paths along BD and AE are
parallel. If riders travel from A to B to D to E, and then
back to A, how far do they travel?
�
_ _ _
2. What is the midsegment triangle in �QRS ?
0
5. Right triangle FGH has midsegments of
length 10 centimeters, 24 centimeters,
and 26 centimeters. What is the area of
�FGH ?
�
1. Name the midsegments in �QRS.
LM; MN; NL
Yes; X is the midpoint of LN, and
2
Identify Relationships
�(4, 8)
_
Holt Geometry
A midsegment triangle is formed from the
three midsegments of a triangle.
6
Y is the midpoint of ML .
Class
Reading Strategies
�
_
Date
A midsegment of a triangle is a segment that joins the midpoints of two sides
of the triangle.
5x � 2
3. Is XY a midsegment of �LMN if its endpoints
are X(8, 2.5) and Y(6.5, �2)? Explain.
32
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Name
The Triangle Midsegment Theorem
A 60 cm
B 120 cm2
4. Find the length of the third side of �ABC
by considering both cases.
�
�
�
�
�
3. How many cases are there to consider
when making a conclusion about the
third side of the triangle? Explain.
�
�
18
12
�
27
11. XJ �
�
Use �ABC for Exercises 4 and 5.
A midsegment of the triangle is 9.
���
��
�
�
2. Find the lengths of the triangle’s sides for each of the cases in Exercise 1.
�
10. m�VXJ �
5-4
�
�
HJ || AC
23
�
�
�
�
��
� Midsegment Thm.
8. VX �
�
�
�
�
�
�
�
�
���
Find each measure.
LESSON
�
�
�
�
_ midsegment connects
Case 2: The
the base EG and one of the
congruent sides of �EFG.
Case 1: The midsegment connects
the two congruent sides EF and FG.
�
_
_
Consider Different Cases
1. Describe two possible cases and make a drawing of each.
�
Given: PQ is a midsegment of �LMN.
_ _
Conclusion: PQ � LN , PQ � _1_ LN
2
JK � _1_ AB
2
4 � _1_ AB
2
8 � AB
Challenge
Triangle EFG is an isosceles triangle
_ with EF � FG and with the perimeter
equal to 22 units. A midsegment, QR , of �EFG is equal to 4 units.
�
You can use the Triangle Midsegment Theorem to
find various measures in �ABC.
HJ � _1_ AC
� Midsegment Thm.
2
HJ � _1_ (12)
Substitute 12 for AC.
2
Simplify.
HJ � 6
Class
When solving a problem, it is sometimes necessary to consider more than
one possible case. It is helpful to make a drawing of each case.
Example
Triangle Midsegment Theorem
A midsegment of a triangle is parallel to
a side of the triangle, and its length is
half the length of that side.
Date
�
�
7. What are the names of the midsegments?
Answers will vary based on
�
_
students’ choice of letters: QR,
_ _
�
RS, QS.
�
�
8. What is the midsegment triangle?
H 90�
J 130�
�QRS
33
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
74
34
Holt Geometry
Holt Geometry