What is an ALTITUDE?
B
Generally “altitude” is another word for height.
o
For a TRIANGLE: a line at right angles to a side that goes through the
opposite corner.
What is the Arithmetic Mean?
(arithmetic series: 1, 4, 7, 10, 13, 16, a, b, c, ….)
x
A
𝒂+𝒄
𝟐
C
= 𝒃 (average of 2 numbers)
The arithmetic mean of 10 & 16 = 13
(geometric series: 4, 12, 36, 108, 324, 972, a, b, c, …)
𝟏𝟐 𝟑𝟔
=
𝟒
𝟏𝟐
12
𝒂
What is the Geometric Mean?
4
=3
36
12
=3
𝒙
The geometric mean of 4 and 36 = 12
What is the geometric mean
of 2 and 8?
=
𝒙
𝒃
x is the geometric mean of a
and b
What is the geometric mean
of 6 and 24?
What is the geometric mean
of 5 and 15?
2. What is the geometric mean
of 10 and 40?
3. What is the geometric mean
of 8 and 16?
NYTS (Now You Try Some)
1. What is the geometric mean
of 9 and 16?
So the question becomes…. What does this have to do with geometry? Great question!! This unique
proportion appears in a special type of triangle and it has a lot to do with similarity.
Geometric Mean Method 1
Use this method if the problem involves the HYPOTENUSE
and one LEG of the large triangle
𝒅
𝒂
=
𝒂 𝒅+𝒆
𝒄𝒍𝒐𝒔𝒆𝒔𝒕 𝒑𝒂𝒓𝒕 𝒐𝒇 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒍𝒆𝒈
=
𝒍𝒆𝒈
𝒘𝒉𝒐𝒍𝒆 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
Geometric Mean Method 2
Use this method if the problem involves the HYPOTENUSE
and the ALTITUDE
𝒅 𝒉
=
𝒉 𝒆
𝒑𝒂𝒓𝒕 𝒐𝒇 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
𝒂𝒍𝒕𝒊𝒕𝒖𝒅𝒆
=
𝒂𝒍𝒕𝒊𝒕𝒖𝒅𝒆
𝒐𝒕𝒉𝒆𝒓 𝒑𝒂𝒓𝒕 𝒐𝒇 𝒉𝒚𝒑𝒐𝒕𝒆𝒏𝒖𝒔𝒆
Let’s apply these geometric mean relationships to solve problems.
x
z
y
9 cm
4 cm
6 cm
z
y
x
10 cm
z
y
5 cm
12 cm
x
x
y
3 cm
z
9 cm