9.3
Altitude-On-Hypotenuse
Theorems
(a.k.a Geometry Mean)
When an altitude is drawn from to a
hypotenuse then three similar triangles
are formed.
Identify the three similar triangles!
Which theorem can they be proven
similar?
Remember: Given two numbers a and
b, x is the geometric mean if
EXTREMES
MEANS
a x

x b
Find the geometric mean of 9 and 16
a x

x b
9 x

x 16
x  144
2
x  12
Altitudes!!!!
What do you remember?
C
A
D
B
An Altitude
*Starts at a vertex
*Forms a right angle
The hypotenuse is split into two pieces
BD and DA
C
A
D
B
CD (the altitude) is the
geometric mean of those
two pieces
If the altitude is in the means place, put the
two segments of the hypotenuse into the
extremes.
C
A
D
B
1st piece hyp
altitude

altitude
2nd piece hyp
If the leg is in the means place, put the whole hyp.
and segment closest to that leg into the extremes.
A
whole hyp
leg

D
leg
side close to leg
the LEG is the
geometric mean
AB
AC
C
B

AC
AD
A
whole hyp
leg

leg
side close to leg
D
AB
CB
C
the LEG is the
geometric mean
B

CB
BD
EX: 1 Find x
C
A
x
9
16
D
9
B
x

x
16
x  144
2
x  12
EX: 2 Find the length of AB.
A

15
D
32
1024
68.26 
x
15
C
B
EX 3: Solve for x
C
A
4
x
D
3
B
16
x
 5.33
3
EX: 4 Find the length of CB.
A
50
D
32
x = 40
C
x
B
EX: 5 Solve for x
C
A
5
12
D
x
B
25
x
 2.08
12