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