obj. 32 similar right triangles
TRANSCRIPT
Similar Right Triangles
The student is able to (I can):
• Use geometric mean to find segment lengths in right triangles
• Apply similarity relationships in right triangles to solve problems.
geometric mean
A proportion with the following pattern:
or or
Example: Find the geometric mean of 5 and 20.
=a x
x b
=2x ab =x ab
=5 x
x 20
=2x 100
x = 10
Theorem The altitude to the hypotenuse of a right triangle forms two triangles that are similar to each other and to the original triangle.
T
I
ME
I T
M
E
I
T
IE
M
90˚−∠M
90˚−∠M
∆MIT ~ ∆IET ~ ∆MEI
Because the triangles are similar, we can set up proportions between the sides:
T
I
ME
I T
M
E
I
T
IE
M
∆MIT ~ ∆IET ~ ∆MEI
= =MI IE ME
, etc.IT ET EI
Corollary The length of the altitude to the hypotenuse is the geometric mean of the lengths of the two segments.
H
E
AT
ab
xy
c
h
=x h
h yor =
2h xy
The length of a leg of a right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse adjacent to that leg.
H
E
AT
ab
xy
c
h
=x a
a cor =
2a xc
=y b
b cor =
2b yc
When you are setting up these problems, remember you are basically setting up similar triangles.
Example
1. Find x, y, and z.
9 6
x
y
z
=9 6
6 x
9x = 36x = 4
=+
9 z
z 9 4
=2z 117=z 3 13
=+
4 y
y 9 4
=2y 52=y 2 13