Triangle Proportionality
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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Printed: May 18, 2016
AUTHORS
Dan Greenberg
Lori Jordan
Andrew Gloag
Victor Cifarelli
Jim Sconyers
Bill Zahner
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C HAPTER
Chapter 1. Triangle Proportionality
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Triangle Proportionality
Here you’ll learn how to apply both the Triangle Proportionality Theorem, which states that if a line that is parallel
to one side of a triangle intersects the other two sides, then it divides those sides proportionally, and its converse.
What if you were given a triangle with a line segment drawn through it from one side to the other? How could you
use information about the triangle’s side lengths to determine if that line segment is parallel to the third side? After
completing this Concept, you’ll be able to answer questions like this one.
Watch This
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/136687
CK-12 Foundation: Triangle Proportionality
First watch this video.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/1354
James Sousa: Triangle Proportionality Theorem
Now watch this video.
MEDIA
Click image to the left or use the URL below.
URL: https://www.ck12.org/flx/render/embeddedobject/1355
James Sousa: Using the Triangle Proportionality Theorem to Solve for Unknown Values
Guidance
Think about a midsegment of a triangle. A midsegment is parallel to one side of a triangle and divides the other two
sides into congruent halves. The midsegment divides those two sides proportionally. But what about another line
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that is parallel, but does not divide the other two sides into congruent halves? In fact, such a line will still divide the
sides proportionally. This is called the Triangle Proportionality Theorem.
Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it
divides those sides proportionally.
If DEkAC, then
BD
DA
=
BE
EC .
DA
( BD
=
EC
BE
is also a true proportion.)
The converse of this theorem is also true.
Triangle Proportionality Theorem Converse: If a line divides two sides of a triangle proportionally, then it is
parallel to the third side.
If
BD
DA
=
BE
EC ,
then DEkAC.
Example A
A triangle with its midsegment is drawn below. What is the ratio that the midsegment divides the sides into?
The midsegment splits the sides evenly. The ratio would be 8:8 or 10:10, which both reduce to 1:1.
Example B
In the diagram below, EBkCD. Find BC.
To solve, set up a proportion.
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Chapter 1. Triangle Proportionality
10 BC
=
−→ 15(BC) = 120
15
12
BC = 8
Example C
Is DEkCB?
If the ratios are equal, then the lines are parallel.
6
8
1
=
=
18 24 3
Because the ratios are equal, DEkCB.
MEDIA
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URL: https://www.ck12.org/flx/render/embeddedobject/136688
CK-12 Foundation: Triangle Proportionality
–>
Guided Practice
Use the diagram to answers questions 1-5. DBkFE.
1. Name the similar triangles. Write the similarity statement.
2.
3.
BE
EC
EC
CB
=
=
?
FC
CF
?
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4.
5.
DB
BC
? = EC
?
FC+?
FC = FE
Answers:
1. 4DBC ∼ 4FEC
2. DF
3. DC
4. FE
5. DF; DB
Explore More
Use the diagram to answer questions 1-7. ABkDE.
1.
2.
3.
4.
5.
6.
7.
Find BD.
Find DC.
Find DE.
Find AC.
What is BD : DC?
What is DC : BC?
Why BD : DC 6= DC : BC?
Use the given lengths to determine if ABkDE.
8.
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Chapter 1. Triangle Proportionality
9.
10.
11.
12.
13.
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 7.8.
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