Obj. 39 Similar Right Triangles
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Obj. 39 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.
description
Use geometric mean to find segment lengths in right triangles. Apply similarity relationships in right triangles to solve problems.
Transcript of Obj. 39 Similar Right Triangles
Obj. 39 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
Thm 8-1-1 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
Cor. 8-1-2 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
Cor. 8-1-3 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
