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3.8.3 Similar Triangle Properties
The student is able to (I can):
• Use properties of similar triangles to find segment lengths.
• Apply proportionality and triangle angle bisector theorems.
• Apply triangle angle bisector theorems
• Use ratios to make indirect measurements
• Use scale drawings to solve problems.
Triangle Proportionality Theorem
If a line parallel to a side of a triangle intersects the other two sides then it divides those sides proportionally.
S
P
A
C
E
>
>
PC SE�
AP AC
PS CE=
Note: This ratio is not the same as the ratio between the third sides!
≠AP PC
PS SE
Triangle Proportionality Theorem Converse
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
S
P
A
C
E
>
>
PC SE�
AP AC
PS CE=
Two Transversal Proportionality
If three or more parallel lines intersect two transversals, then they divide the transversals proportionally.
G
O
D
T
A
C>
>
>
CA DO
AT OG=
Examples Find PE
10x = (4)(14)
10x = 56
S
C
O
P
E
10101010 14141414
4444
10 14
4 x=
xxxx
28 3x 5 5.6
5 5= = =
>
>
Example Verify that
(15)(8) = (10)(12)?
120 = 120 � Therefore,
H
O
RSE
HE OS�
15
10
12 8
=15 10
?12 8
HE OS�
Example Solve for x.
6x = (10)(9)
6x = 90
x = 15
>
>
>
x
96
10
10 x
6 9=
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.
=CD CA
DB AB
Example: Solve for x.
=AD AB
DC BC
=
=
= =
3.5 5
x 125x 42
42x 8.4
5
indirect measurement
Any method that uses formulas, similar figures, and/or proportions to measure an object.
Example: An 8 foot tall stick casts a 6 foot shadow. At the same time, a tall flagpole casts an 18 foot shadow. How tall is the flagpole?
6
8
18
x
The triangles are similar by AA~.
8 x
6 18= 6x = 144 → x = 24 feet
Example Miriam saw a mirror on the ground and noticed that she could see the top of Reunion Tower in the mirror. Her line of sight was 5’ above the ground, and the mirror was 2’ away from her. She measured the distance from that position to the base of Reunion Tower, and it was 224 feet. How high is Reunion Tower?
The reflection creates congruent angles, so the triangles are similar by AA~.
Example
5’
2’ 224’
x
5 x
2 2242x 1120
x 560 ft.
=
=
=
