7.4 Triangle Proportionality Theorems

Use properties of similar triangles to solve problems.

1. Proportionality Theorems
The student is able to (I can):
• Use properties of similar triangles to find segmen t lengths.
• Apply proportionality and triangle angle bisector
theorems.
• Apply triangle angle bisector theorems

2. Triangle Proportionality Theorem
If a line parallel to a side of a triangle intersec ts the other
two sides then it divides those sides proportionall y.
S
P
A
C
E
>
>
PC SE
S
AP AC PS CE
=
This ratio is notthe same as the ratio between the third sides!
AP PC PS SE
≠

3. Triangle Proportionality Theorem Converse
If a line divides two sides of a triangle proportio nally, then
it is parallel to the third side.
S
P
A
C
E
>
>
PC SE
S
AP AC PS CE
=

4. Two Transversal Proportionality
If three or more parallel lines intersect two trans versals,
then they divide the transversals proportionally.
G
O
D
T
A
C
>
>
>
CA DO AT OG
=

5. Example:
Find PE
S
C
O
P
E
1014
4
x
>
>

6. Example:
Find PE
10x= (4)(14)
10x= 56
S
C
O
P
E
1014
4
10 14
4
x
=
x
28 3
5 5.6
5 5
x
= = =>
>

7. Example:
Verify that
H
O
R
HE OS
S
15
10

8. Example:
Verify that
(15)(8) = (10)(12)?
120 = 120 STherefore,
H
O
R
S E
HE OS
S
15
10
12 8
15 10
?
12 8
=
HE OS
S

9. Example:
Solve for x.
>
>
>
x
9 6
10

10. Example:
Solve for x.
6x = (10)(9)
6x = 90
x = 15
>
>
>
x
9 6
10
10
6 9
x
=

11. Triangle Angle Bisector Theorem An angle bisector of an angle of a triangle divides the
opposite side in two segments that are proportional to the
other two sides of the triangle.
bisects ∠CAB
or
CD CA CD DB DB AB CA AB
= =
DA

12. Example: Solve for x.

13. Example: Solve for x.
3.5
5 12
5 42
42
8.4
5
AD DC AB BC
x
x
x
=
=
=
= =