Download 9-3: Altitude-on-Hypotenuse Theorems

7-4: Similarity in Right Triangles
Goal: Be able to find and use relationships in
smaller right triangles
geometric mean - The number x such that
a x

x b
where a, b, and x are positive numbers.
Ex 1: Find the geometric mean of
a.) 4 and 7
b.) 5 and 25
4 x

x 7
5 x

x 25
x 2  28
x 2  125
x2 7
x5 5
Similarity in Right Triangles
When _____________
altitude
hypotenuse
NO is drawn to the ________________
of
PMN , 3 ______________
similar
triangles are formed.
M
x
O
a
y
h
N
b
P
1
c
x
a
3
y
h
b
h
b
a
x
b
c
y
h
2
a
NOM ~ PON ~ PNM
M
P
c
x
P
O
N
a
h
y
h
O
N
b
NOM ~ PON
PON
NOM
x h
= so h2=xy
h y
b
y
a
x
c
b
M
O
h
a
N
N
M
P
NOM ~ PNM
PNM
PON~PNM
PNM NOM
PON
y b
= so b2=cy
b c
h is the geometric mean b is the geometric mean
between x and y
between y and c
x a
= so a2=cx
a c
a is the geometric mean
between x and c
Ex 3: Find x.
Ex 4: Find x.
x
x
3
5
3 x

x 5
x 2  15
x  15
5
7
5 x

x 7
x 2  35
x  35
Ex 5: Find x.
Ex 6: Find x.
x
3
2
x
y
6
5
x 6

6 2
3 x

x 8
2x  36
x  24
x  18
2
x2 6
182  62  y 2
360  y 2
6 10  y
Ex 7: Find x.
Ex 8: Find x.
3
z
x
y
2 6
y
1
1 x

x 5
x 5
2
x 5
z
4
1 y

y 4
y2  4
y2
x
4 z

z 5
3
2 6

2 6 3 y
5 x

x 8
5 z

z 3
2
2
2
2
z
 15
(2)
(
6)

3(3

y
)
z  20
x  40
24  9  3y
x  2 10 z  15
z2 5
15  3y
2
5 y