Download The altitude to the hypotenuse of a right triangle forms two triangles

8.1 Similar Right Triangles
With a 3 x 5 card, use a straightedge to form 3 right
triangles as seen below. Cut the three triangles out.
B
C
C
A
D
D
A
B
C
Theorem: The altitude to the hypotenuse of
a right triangle forms two triangles that are
similar to each other and to the original
triangle.
C
A
B
D
ADC ~
ACB ~
CDB
Now pull your triangles apart and arrange them so
corresponding parts are in the same position:
B
B
C
A
D
C
A
C
ADC ~
(1 & 2)
AD
AC
=
AC
D
ACB ~
(1 & 3)
AD CD
=
CD
CDB
(2 & 3)
AB =
CB
CB
Look at the first proportion we made:
AD
AC
=
AC
AB
The means of the proportion are the same number.
That number is the geometric mean of the extremes.
The geometric mean of two positive numbers
is the positive square root of their product.
a
x
=
x
b
Find the geometric mean of 2 and 8
Find the geometric mean of 10 and 30
Corollary: The length of the altitude to the hypotenuse of a
right triangle is the geometric mean of the lengths of the two
segments of the hypotenuse.
x
a
c
y
h
h
2
= xy
b
Corollary: 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.
x
a
c
a
2
= xc
y
h
b
b
2
= yc
1.
x
6
10
2.
5
y
8
3.
h
16
5.75
4.
z
y
x
4
9
5.
x
9
15
y
6.
25
w
z
y
9
x