Page 1 of 7
7.5
Goal
Use the Triangle
Proportionality Theorem
and its converse.
Proportions and
Similar Triangles
Geo-Activity
Investigating Proportional Segments
1 Draw a triangle. Label its vertices ●
2 Draw a line through D parallel
●
&*. Label the intersection of
to AC
the line and BC
&* as point E.
A, B, and C. Make sure that each
side is at least 4 cm. Draw a point
on AB
&*. Label the point D.
Key Words
• midsegment of a triangle
B
B
D
D
E
A
A
C
C
3 Measure BD
&*, DA
&*, BE
&*, and EC
&* in centimeters. Then calculate the
●
BE
EC
BD
DA
ratios }} and }} .
4 Make a conjecture about the ratios of segment lengths of a
●
triangle’s sides when the triangle is cut by a line parallel to the
triangle’s third side.
&** and a point Q lies on
Proportionality Suppose that a point P lies on GH
GP
JQ
&*. If }} 5 }} , then we say that GH
&* and &*
JK
JK are divided proportionally.
PH
QK
G
3
P
6
H
J
4
P
K
8
&* divides AB
&* and CB
&* proportionally.
In the Geo-Activity above, DE
THEOREM 7.4
Triangle Proportionality Theorem
Words
If a line parallel to one side of
a triangle intersects the other
two sides, then it divides the
two sides proportionally.
Symbols
386
Chapter 7
Similarity
R
T
P
RT
RU
In T QRS, if TU
&* i QS
&*, then }} 5 }} .
TQ
US
U
S
Page 2 of 7
EXAMPLE
Find Segment Lengths
1
Find the value of x.
C
4
D
x
E
12
8
B
A
Solution
CD
CE
}} 5 } }
DB
EA
Triangle Proportionality Theorem
4
x
}} 5 } }
8
12
Substitute 4 for CD, 8 for DB, x for CE, and 12 for EA.
4 p 12 5 8 p x
Cross product property
48 5 8x
Multiply.
48
8x
}} 5 }}
8
8
Divide each side by 8.
65x
IStudent Help
EXAMPLE
Simplify.
2
Find Segment Lengths
ICLASSZONE.COM
MORE EXAMPLES
More examples at
classzone.com
Find the value of y.
R
9
P
S
T
y
3
P
20
Solution
You know that PS 5 20 and PT 5 y. By the Segment Addition
Postulate, TS 5 20 2 y.
PT
PQ
}} 5 } }
TS
QR
y
3
}} 5 }
9
20 2 y
3(20 2 y) 5 9 p y
Triangle Proportionality Theorem
Substitute 3 for PQ, 9 for QR,
y for PT, and (20 2 y) for TS.
Cross product property
60 2 3y 5 9y
Distributive property
60 2 3y 1 3y 5 9y 1 3y
Add 3y to each side.
60 5 12y
Simplify.
60
12y
}} 5 } }
12
12
Divide each side by 12.
55y
Simplify.
7.5
Proportions and Similar Triangles
387
Page 3 of 7
THEOREM 7.5
Converse of the Triangle Proportionality Theorem
Words
If a line divides two sides of a triangle
proportionally, then it is parallel to
the third side.
Symbols
R
T
RT
RU
In TQRS, if }} 5 }} , then TU
&* i QS
&*.
TQ
US
EXAMPLE
U
P
S
Determine Parallels
3
Given the diagram, determine whether
xxxxxx is parallel to GH
xxxxx.
MN
G
21
M
56
L
N 16 H
48
Solution
xxxxxx.
Find and simplify the ratios of the two sides divided by MN
LN
3
48
}} 5 }} 5 }}
NH
1
16
LM
56
8
}} 5 } } 5 } }
MG
21
3
ANSWER
8
3
xxxxxx is not parallel to GH
xxxxx.
© Because }3} Þ }1}, MN
Find Segment Lengths and Determine Parallels
Find the value of the variable.
4
1.
2.
x
5
14
15
y
6
10
&* is parallel to ST
&*. Explain.
Given the diagram, determine whether QR
T
3.
S
21
23
R
P
4. P
15
17
P
4
6
P
R
12
8
S
388
Chapter 7
Similarity
T
Page 4 of 7
A midsegment of a triangle is a segment that connects the midpoints
of two sides of a triangle. The following theorem about midsegments is
a special case of the Triangle Proportionality Theorem.
THEOREM 7.6
The Midsegment Theorem
Words
The segment connecting the midpoints
of two sides of a triangle is parallel to
the third side and is half as long.
C
E
D
Symbols
In T ABC, if CD 5 DA and CE 5 EB,
1
&* i AB
&* and DE 5 }} AB.
then DE
B
A
2
EXAMPLE
Use the Midsegment Theorem
4
&*.
Find the length of QS
R
P
S
P
Solution
T
10
From the marks on the diagram, you know S is the midpoint of
&*, and Q is the midpoint of RP
&*. Therefore, QS
&* is a midsegment
RT
of T PRT. Use the Midsegment Theorem to write the following
equation.
1
2
1
2
QS 5 }}PT 5 }}(10) 5 5
ANSWER
&* is 5.
© The length of QS
Use the Midsegment Theorem
Find the value of the variable.
5.
q
6.
8
p
11
14
11
8
16
7. Use the Midsegment Theorem to find
A
the perimeter of T ABC.
3
3
B
7.5
5
4
4
C
Proportions and Similar Triangles
389
Page 5 of 7
7.5 Exercises
Guided Practice
Vocabulary Check
Complete the statement.
1. The __?__ Theorem states that if a line divides two sides of a
triangle proportionally, then it is __?__ to the third side.
2. A __?__ of a triangle is a segment that connects the midpoints
of two sides of a triangle.
Skill Check
Copy and complete the proportion using the diagram below.
3. }} 5 }}
?
EC
4. }} 5 }}
AD
AE
5. }} 5 }}
?
AC
BD
CE
6. }} 5 }}
BA
?
AD
DB
?
DA
CE
EA
A
D
E
B
C
Find the value of the variable.
7.
4
8.
x
8
z
9.
10
4
y
15
3
2
Practice and Applications
Extra Practice
Using Algebra Solve the proportion.
See p. 688.
2
3
m
36
t
2
10. }} 5 }}
5
12
7
18
21
y
11. }} 5 }}
3
4
27
r
12. }} 5 }}
13. }} 5 }}
Finding Segment Lengths Find the value of the variable.
14.
15.
9
7
16.
21
y
Example 1:
Example 2:
Example 3:
Example 4:
390
Exs. 14–19
Exs. 14–19
Exs. 20–23
Exs. 24–29,
33–37
Chapter 7
Similarity
17.
5
18.
q
6
6
7
Homework Help
p
18
x
20
15
12
20
19.
z
55
15
24
24
c
12
32
Page 6 of 7
&* is
Determining Parallels Given the diagram, determine whether QS
&*.
parallel to PT
R
20.
21.
16
4
8
P
12.5
S
2
T
P
R
T
22.
34
P
P
35
16
R
P
R
12
P
9
15
P 4
10
23.
S
T
5
S
20
S
15
T
P
Using the Midsegment Theorem Find the value of the variable.
a
24.
25.
26.
18
11
x
b
4
7
27.
Fractals
7
5
c
28.
29.
y
8
8
10
19
10
z
19
27
1
Visualize It! The design below approximates a fractal. Begin with
an equilateral triangle. Shade the triangle formed by the three
midsegments. Continue the process for each unshaded triangle.
16
16
FRACTALS are shapes that
look the same at many levels
of magnification. Take a small
part of the image above and
you will see that it looks
similar to the whole image.
Application Links
CLASSZONE.COM
16
Stage 0
Stage 1
Stage 2
Stage 3
30. Find the perimeter of the dark blue triangle in Stage 1.
31. Challenge Find the total perimeter of all the dark blue triangles
in Stage 2.
32. Challenge Find the total perimeter of all the dark blue triangles
in Stage 3.
7.5
Proportions and Similar Triangles
391
Page 7 of 7
Midsegment Theorem Use the diagram below to complete
the statement.
B
xxxxx i __?__
&* i __?__
33. LM
34. AB
35. If AC 515, then LN 5 __?__.
L
N
36. If MN 5 7.4, then AB 5 __?__.
37. If NC 5 9.5, then LM 5 __?__.
A
M
C
Technology In Exercises 38 and 39, use geometry software to
complete the steps below.
1 Draw T ABC.
●
2 Construct the angle bisector of aA.
●
3 Construct the intersection of the
●
B
D
&*. Label it D.
angle bisector and BC
A
&**, AC
&*, DB
&*, and AB
&*. Then
4 Measure DC
●
C
DC
DB
calculate the ratios }} and }}.
AC
AB
38. Drag one or more of the triangle’s vertices. What do you notice
about the ratios as the shape changes?
39. Complete the conjecture: If a ray bisects an angle of a triangle,
then it divides the opposite side into segments whose lengths are
__?__ to the lengths of the other two sides.
Standardized Test
Practice
40. Multiple Choice What is the value of x?
A
X
C
X
21
32
B 24
X
D 42
X
8
x
42
Mixed Review
Reflections Determine if the entire word has any lines of symmetry.
If so, write the word and draw the line(s) of symmetry. (Lesson 5.7)
41.
Algebra Skills
392
Chapter 7
Similarity
16
42.
43.
Finding Slope Find the slope of the line that passes through the
points. (Skills Review, p. 665)
44. (0, 2) and (4, 8)
45. (1, 2) and (3, 4)
46. (5, 2) and (5, 3)
47. (25, 6) and (1, 2)
48. (24, 4) and (2, 0)
49. (3, 7) and (21, 23)
50. (5, 3) and (21, 1)
51. (0, 24) and (3, 5)
52. (23, 2) and (6, 25)