8.6 Proportions

1
8.6 Proportions Similar Triangles
2
Objectives/Assignments

• Use proportionality theorems to calculate segment
  lengths
• Assignment 2-34 even

3
Use Proportionality Theorems

• In this lesson, you will study four
  proportionality theorems. Similar triangles are
  used to prove each theorem.

4
Theorems 8.4 Triangle Proportionality Theorem
If a line parallel to one side of a triangle
intersects the other two sides, then it divides
the two side proportionally. If TU QS, then
RT
RU

TQ
US
5
Theorems 8.5 Converse of the Triangle
Proportionality Theorem
If a line divides two sides of a triangle
proportionally, then it is parallel to the third
side.
RT
RU
If
, then TU QS.

TQ
US
6
Ex. 1 Finding the length of a segment

• In the diagram AB ED, BD 8, DC 4, and AE
  12. What is the length of EC?

7

• Step
• DC EC
• BD AE
• 4 EC
• 8 12
• 4(12)
• 8
• 6 EC

• Reason
• Triangle Proportionality Thm.
• Substitute
• Multiply each side by 12.
• Simplify.

EC

• So, the length of EC is 6.

8
Ex. 2 Determining Parallels

• Given the diagram, determine whether MN GH.

LM
56
8


MG
21
3
LN
48
3


NH
16
1
8
3
?
3
1
MN is not parallel to GH.
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Theorem 8.6

• If three parallel lines intersect two
  transversals, then they divide the transversals
  proportionally.
• If r s and s t and l and m intersect, r, s,
  and t, then

UW
VX

WY
XZ
10
Theorem 8.7

• 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.
• If CD bisects ?ACB, then

AD
CA

DB
CB
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Ex. 3 Using Proportionality Theorems

• In the diagram ?1 ? ?2 ? ?3, and PQ 9, QR
  15, and ST 11. What is the length of TU?

12
SOLUTION Because corresponding angles are
congruent, the lines are parallel and you can use
Theorem 8.6
PQ
ST
Parallel lines divide transversals proportionally.

QR
TU
9
11

Substitute
15
TU
9 ? TU 15 ? 11 Cross Product property
15(11)
55

TU

Divide each side by 9 and simplify.
9
3

• So, the length of TU is 55/3 or 18 1/3.

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Ex. 4 Using the Proportionality Theorem

• In the diagram, ?CAD ? ?DAB. Use the given side
  lengths to find the length of DC.

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Solution

• Since AD is an angle bisector of ?CAB, you can
  apply Theorem 8.7. Let x DC. Then BD 14 x.

AB
BD

Apply Thm. 8.7
AC
DC
9
14-X
Substitute.

15
X
15
Ex. 4 Continued . . .
9 ? x 15 (14 x) 9x 210 15x 24x
210 x 35/4 8.75
Cross product property Distributive Property Add
15x to each side Divide each side by 24.

• So, the length of DC is 8.75 units.

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Use proportionality Theorems in Real Life

• Example 5 Finding the length of a segment
• Building Construction You are insulating your
  attic, as shown. The vertical 2 x 4 studs are
  evenly spaced. Explain why the diagonal cuts at
  the tops of the strips of insulation should have
  the same length.

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Use proportionality Theorems in Real Life

• Because the studs AD, BE and CF are each
  vertical, you know they are parallel to each
  other. Using Theorem 8.6, you can conclude that

DE
AB

EF
BC

• Because the studs are evenly spaced, you know
  that DE EF. So you can conclude that AB BC,
  which means that the diagonal cuts at the tops of
  the strips have the same lengths.

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Ex. 6 Finding Segment Lengths

• In the diagram KL MN. Find the values of the
  variables.

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Solution

• To find the value of x, you can set up a
  proportion.

9
37.5 - x

Write the proportion Cross product
property Distributive property Add 13.5x to each
side. Divide each side by 22.5
13.5
x
13.5(37.5 x) 9x 506.25 13.5x 9x
506.25 22.5 x 22.5 x

• Since KL MN, ?JKL ?JMN and

JK
KL

JM
MN
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Solution

• To find the value of y, you can set up a
  proportion.

9
7.5

Write the proportion Cross product
property Divide each side by 9.
13.5 9
y
9y 7.5(22.5)
y 18.75