8.4 Proportionality Theorems Essential Question

8.4
TEXAS ESSENTIAL
KNOWLEDGE AND SKILLS
Proportionality Theorems
Essential Question
What proportionality relationships exist in a
triangle intersected by an angle bisector or by a line parallel to one of the sides?
G.8.A
Discovering a Proportionality Relationship
Work with a partner. Use dynamic geometry software to draw any △ABC.
— parallel to BC
— with endpoints on AB
— and AC
—, respectively.
a. Construct DE
B
D
C
E
ANALYZING
MATHEMATICAL
RELATIONSHIPS
To be proficient in math,
you need to look closely
to discern a pattern
or structure.
A
b. Compare the ratios of AD to BD and AE to CE.
— to other locations parallel to BC
— with endpoints on AB
— and AC
—,
c. Move DE
and repeat part (b).
d. Change △ABC and repeat parts (a)–(c) several times. Write a conjecture that
summarizes your results.
Discovering a Proportionality Relationship
Work with a partner. Use dynamic geometry software to draw any △ABC.
a. Bisect ∠B and plot point D at the
intersection of the angle bisector
—.
and AC
b. Compare the ratios of AD to DC
and BA to BC.
B
A
D
C
c. Change △ABC and repeat parts (a)
and (b) several times. Write a
conjecture that summarizes
your results.
Communicate Your Answer
B
3. What proportionality relationships exist in a triangle
intersected by an angle bisector or by a line parallel
to one of the sides?
4. Use the figure at the right to write a proportion.
Section 8.4
D
E
A
Proportionality Theorems
C
449
8.4 Lesson
What You Will Learn
Use the Triangle Proportionality Theorem and its converse.
Use other proportionality theorems.
Core Vocabul
Vocabulary
larry
Using the Triangle Proportionality Theorem
Previous
corresponding angles
ratio
proportion
Theorems
Theorem 8.6 Triangle Proportionality Theorem
Q
If a line parallel to one side of a triangle
intersects the other two sides, then it divides
the two sides proportionally.
T
R
U
S
RT
RU
If —
TU —
QS , then — = —.
TQ
US
Proof Ex. 27, p. 455
Theorem 8.7 Converse of the Triangle Proportionality Theorem
Q
If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
T
R
U
S
RT
RU
If — = —, then TU QS.
TQ
US
Proof Ex. 28, p. 455
Finding the Length of a Segment
— UT
—, RS = 4, ST = 6, and QU = 9. What is the length of RQ
—?
In the diagram, QS
R
4
Q
S
9
6
U
T
SOLUTION
—=—
RQ
QU
RS
ST
Triangle Proportionality Theorem
—=—
RQ
9
4
6
Substitute.
RQ = 6
Multiply each side by 9 and simplify.
— is 6 units.
The length of RQ
Monitoring Progress
1.
—.
Find the length of YZ
Help in English and Spanish at BigIdeasMath.com
V
35
W
44
Y
Z
450
Chapter 8
Similarity
X
36
The theorems on the previous page also imply the following:
Contrapositive of the Triangle
Proportionality Theorem
RT RU
— QS
—.
If — ≠ —, then TU
TQ US
Inverse of the Triangle
Proportionality Theorem
RT RU
— QS
—, then —
≠ —.
If TU
TQ US
Solving a Real-Life Problem
On the shoe rack shown, BA = 33 centimeters,
CB = 27 centimeters, CD = 44 centimeters, and
DE = 25 centimeters. Explain why the shelf is not
parallel to the floor.
C
B
D
A
SOLUTION
E
Find and simplify the ratios of the lengths.
CD
DE
44
CB 27
9
—=—=—
25
BA 33 11
9 —
44
—. So, the shelf is not parallel to the floor.
Because — ≠ —, BD is not parallel to AE
25 11
—=—
P
Q
50
Monitoring Progress
90
N
72
Help in English and Spanish at BigIdeasMath.com
— QR
—.
2. Determine whether PS
S 40 R
Recall that you partitioned a directed line segment in the coordinate plane in Section
3.5. You can apply the Triangle Proportionality Theorem to construct a point along a
directed line segment that partitions the segment in a given ratio.
Constructing a Point along
a Directed Line Segment
— so that the ratio of AL to LB is 3 to 1.
Construct the point L on AB
SOLUTION
Step 1
Step 2
Step 3
C
C
C
G
G
F
F
E
E
D
A
B
Draw a segment and a ray
— of any length. Choose any
Draw AB
AB. Draw ⃗
AC.
point C not on ⃖⃗
A
D
B
Draw arcs Place the point of
a compass at A and make an arc
of any radius intersecting ⃗
AC . Label
the point of intersection D. Using the
same compass setting, make three
more arcs on ⃗
AC, as shown. Label the
points of intersection E, F, and G and
note that AD = DE = EF = FG.
Section 8.4
A
J
K
L
B
—. Copy ∠AGB
Draw a segment Draw GB
and construct congruent angles at D, E, and
— at J, K, and L.
F with sides that intersect AB
—
—
—
Sides DJ , EK , and FL are all parallel,
— equally. So,
and they divide AB
AJ = JK = KL = LB. Point L divides
directed line segment AB in the ratio 3 to 1.
Proportionality Theorems
451
Using Other Proportionality Theorems
Theorem
Theorem 8.8 Three Parallel Lines Theorem
If three parallel lines intersect two transversals,
then they divide the transversals proportionally.
r
s
t
U
W
Y
V
X
Z
m
UW
WY
VX
XZ
—=—
Proof Ex. 32, p. 455
Using the Three Parallel Lines Theorem
In the diagram, ∠1, ∠2, and ∠3 are all congruent,
GF = 120 yards, DE = 150 yards, and
CD = 300 yards. Find the distance HF between
Main Street and South Main Street.
1
Main St.
120 yd
2
G
SOLUTION
150 yd
D Second St.
3
H
South Main St.
C
Use the Three Parallel Lines Theorem to set up a proportion.
—=—
HG
GF
CD
DE
Three Parallel Lines Theorem
—=—
HG
120
300
150
Substitute.
HG = 240
Multiply each side by 120 and simplify.
By the Segment Addition Postulate (Postulate 1.2),
HF = HG + GF = 240 + 120 = 360.
The distance between Main Street and South Main Street is 360 yards.
Method 2
Set up a proportion involving total and partial distances.
Step 1 Make a table to compare the distances.
Total distance
⃖⃗
CE
⃖⃗
HF
CE = 300 + 150 = 450
HF
DE = 150
GF = 120
Partial distance
Step 2 Write and solve a proportion.
450
150
HF
120
—=—
Write proportion.
360 = HF
Multiply each side by 120 and simplify.
The distance between Main Street and South Main Street is 360 yards.
452
Chapter 8
Similarity
E
300 yd
Corresponding angles are congruent, so
⃖⃗
FE, ⃖⃗
GD, and ⃖⃗
HC are parallel. There are
different ways you can write a proportion
to find HG.
Method 1
F
Theorem
Theorem 8.9 Triangle Angle Bisector Theorem
If a ray bisects an angle of a triangle, then
it divides the opposite side into segments
whose lengths are proportional to the
lengths of the other two sides.
A
D
C
B
AD CA
—=—
DB
CB
Proof Ex. 35, p. 456
Using the Triangle Angle Bisector Theorem
—.
In the diagram, ∠QPR ≅ ∠RPS. Use the given side lengths to find the length of RS
Q
7
P
R
13
15
x
S
SOLUTION
Because ⃗
PR is an angle bisector of ∠QPS, you can apply the Triangle Angle Bisector
Theorem. Let RS = x. Then RQ = 15 − x.
RQ PQ
RS
PS
15 − x
7
—=—
x
13
195 − 13x = 7x
—=—
Triangle Angle Bisector Theorem
Substitute.
Cross Products Property
9.75 = x
Solve for x.
— is 9.75 units.
The length of RS
Monitoring Progress
Help in English and Spanish at BigIdeasMath.com
Find the length of the given line segment.
—
3. BD
40
16
A
E
—
4. JM
M
3
C
2
G
1
B D
16 J
15
H
F
30
18
K
N
Find the value of the variable.
5.
S
24
V
14
6.
Y
4
T
x
48
4 2
U
Section 8.4
W
Z
4
y
X
Proportionality Theorems
453
Exercises
8.4
Dynamic Solutions available at BigIdeasMath.com
Vocabulary and Core Concept Check
1. COMPLETE THE STATEMENT If a line divides two sides of a triangle proportionally, then it is
_________ to the third side. This theorem is known as the ___________.
— and ⃗
2. VOCABULARY In △ABC, point R lies on BC
AR bisects ∠CAB. Write the proportionality
statement for the triangle that is based on the Triangle Angle Bisector Theorem (Theorem 8.9).
Monitoring Progress and Modeling with Mathematics
—.
In Exercises 3 and 4, find the length of AB
(See Example 1.)
3.
E 4.
A
A
14
E
12
B
3
C
4
In Exercises 13–16, use the diagram to complete the
proportion.
D
B
12
18
D
F
D
B
G
E
C
C
BD
BF
— || JN
—.
In Exercises 5–8, determine whether KM
BF
DF
CG
13. — = —
14. — = —
DF
16. — = —
CG
(See Example 2.)
5.
6.
L
12
8
K
5
J
J
N
22.5
M
7.5
25
L
N
7.
L
18
M
20
K
8.
M
K
10
N
In Exercises 17 and 18, find the length of the indicated
line segment. (See Example 3.)
—
17. VX
—
18. SU
Y
N
18
J
CG
CE
BD
J
24
15
EG
CE
15. — = —
35
34
K
16
M
15
L
Z
20
W
8
U
15
P 8 R T
12
X
N S
10
V
U
In Exercises 19–22, find the value of the variable.
(See Example 4.)
19.
CONSTRUCTION In Exercises 9–12, draw a segment with
20.
8
the given length. Construct the point that divides the
segment in the given ratio.
z
3
y
1.5
4.5
4
6
9. 3 in.; 1 to 4
21.
10. 2 in.; 2 to 3
16.5
11. 12 cm; 1 to 3
11
12. 9 cm; 2 to 5
454
Chapter 8
p
Similarity
22.
16
q
36
28
29
23. ERROR ANALYSIS Describe and correct the error in
29. MODELING WITH MATHEMATICS The real estate term
solving for x.
✗
lake frontage refers to the distance along the edge of a
piece of property that touches a lake.
x
A
D
14
C
lake
10
16
174 yd
B
AB CD
=
—
BC —
AD
Lot A
10 14
=—
—
x
16
10x = 224
x = 22.4
48 yd
a. Find the lake frontage (to the nearest tenth) of
each lot shown.
b. In general, the more lake frontage a lot has, the
higher its selling price. Which lot(s) should be
listed for the highest price?
the student’s reasoning.
B
c. Suppose that lot prices are in the same ratio as lake
frontages. If the least expensive lot is $250,000,
what are the prices of the other lots? Explain
your reasoning.
D
A
Lot C
61 yd
55 yd
Lakeshore Dr.
24. ERROR ANALYSIS Describe and correct the error in
✗
Lot B
C
BD AB
Because — = — and BD = CD,
CD AC
it follows that AB = AC.
30. USING STRUCTURE Use the diagram to find the
values of x and y.
2
MATHEMATICAL CONNECTIONS In Exercises 25 and 26,
— RS
—.
find the value of x for which PQ
25.
P 2x + 4
S 5
R
Q
26.
T
P
12
R
2x − 2
T
7
3x + 5
1.5
5
Q
x
3
21
y
S
3x − 1
31. REASONING In the construction on page 451, explain
why you can apply the Triangle Proportionality
Theorem (Theorem 8.6) in Step 3.
27. PROVING A THEOREM Prove the Triangle
Proportionality Theorem (Theorem 8.6).
— TU
—
Given QS
32. PROVING A THEOREM Use the diagram with the
Q
T
QT SU
Prove — = —
TR UR
R
U
S
auxiliary line drawn to write a paragraph proof of
the Three Parallel Lines Theorem (Theorem 8.8).
Given k1 k2 k3
Prove
28. PROVING A THEOREM Prove the Converse of the
CB
BA
DE
EF
—=—
Triangle Proportionality Theorem (Theorem 8.7).
ZY
ZX
Given — = —
YW XV
Prove
C
W Y
— WV
—
YX
B
Z
V
X
A
t1
D
t2
E
F
auxiliary
line
k1
k2
k3
Section 8.4
Proportionality Theorems
455
33. CRITICAL THINKING In △LMN, the angle bisector of
—. Classify △LMN as specifically as
∠M also bisects LN
possible. Justify your answer.
38. MAKING AN ARGUMENT Two people leave points A
and B at the same time. They intend to meet at
point C at the same time. The person who leaves
point A walks at a speed of 3 miles per hour. You and
a friend are trying to determine how fast the person
who leaves point B must walk. Your friend claims you
—. Is your friend correct?
need to know the length of AC
Explain your reasoning.
34. HOW DO YOU SEE IT? During a football game,
the quarterback throws the ball to the receiver. The
receiver is between two defensive players, as shown.
If Player 1 is closer to the quarterback when the ball
is thrown and both defensive players move at the
same speed, which player will reach the receiver
first? Explain your reasoning.
A
B
0.9 mi
0.6 mi
D
C
E
39. CONSTRUCTION Given segments with lengths r, s,
r t
and t, construct a segment of length x, such that — = —.
s x
r
s
35. PROVING A THEOREM Use the diagram with the
auxiliary lines drawn to write a paragraph proof of
the Triangle Angle Bisector Theorem (Theorem 8.9).
t
40. PROOF Prove Ceva’s Theorem: If P is any point
⋅ ⋅
AY CX BZ
inside △ABC, then — — — = 1.
YC XB ZA
Given ∠YXW ≅ ∠WXZ
YW XY
Prove — = —
WZ XZ
N
Y
X
A
auxiliary lines
B
Z
M
X
P
W
Z
A
Y
C
— through A and C,
(Hint: Draw segments parallel to BY
as shown. Apply the Triangle Proportionality Theorem
(Theorem 8.6) to △ACM. Show that △APN ∼ △MPC,
△CXM ∼ △BXP, and △BZP ∼ △AZN.)
36. THOUGHT PROVOKING Write the converse of the
Triangle Angle Bisector Theorem (Theorem 8.9).
Is the converse true? Justify your answer.
37. REASONING How is the Triangle Midsegment Theorem
(Theorem 6.8) related to the Triangle Proportionality
Theorem (Theorem 8.6)? Explain your reasoning.
Maintaining Mathematical Proficiency
Reviewing what you learned in previous grades and lessons
Use the triangle. (Section 5.5)
41. Which sides are the legs?
a
42. Which side is the hypotenuse?
c
b
Solve the equation. (Skills Review Handbook)
43. x2 = 121
456
Chapter 8
44. x2 + 16 = 25
Similarity
45. 36 + x2 = 85