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7-4: Similarity in Right TrianglesGoal: Be able to find and use relationships in smaller right triangles
geometric mean -
The number x such that
where a, b, and x are positive numbers.Ex 1: Find the geometric mean of
a.) 4 and 7 b.) 5 and 25
4
7
x
x
a
x b
x
2 28x
2 7x
5
25
x
x
2 125x
5 5x
Similarity in Right TrianglesNO
PMN
b
a
x
y
h
O
N P
M
When _____________ is drawn to the ________________
of , 3 ______________ triangles are formed.
altitude hypotenuse
similar
b
a
cx
yh
12
3
h
yb
a
bc
h
x
a
~ ~NOM PON PNM
b
a
cx
yh
O
N P
M
NOM PONx
h=
h
y so h2=xy
PON PNMy
b=
b
c so b2=cy
NOM PNMx
a=
a
c so a2=cx
h is the geometric mean between x and y
b is the geometric mean between y and c
a is the geometric mean between x and c
~NOM PON ~PON PNM ~NOM PNM
h
yb
a
bc
h
x
a
N
NO OM MN
P
P
Ex 3: Find x.
3
x
5
Ex 4: Find x.
7
x
5
3
5
x
x
2 15x
15x
5
7
x
x
2 35x
35x
Ex 5: Find x.
3
x5
Ex 6: Find x.
6
x
y
3
8
x
x
2 24x
2 6x
2
6
6 2
x
2 36x
18x
2 2 218 6 y
2360 y
6 10 y
Ex 7: Find x.
1
y
4
Ex 8: Find x.
zy
3
x z
1
5
x
x
2 5x
5x
1
4
y
y
2 4y
2y
4
5
z
z
2 20z
2 5z
x
2 6
3 2 6
32 6 y
2 2(2) ( 6) 3(3 )y 24 9 3y 15 3y
5 y
5
8
x
x
2 40x
2 10x
5
3
z
z
2 15z
15z
