Right Triangle Similarity
Geometry Lesson 9.3
Geometric Mean
Consider the proportion . In this case, the �means of the proportion are the same number, and �that number is the geometric mean of the extremes.
The geometric mean of two positive numbers is the positive square root of their product. So the geometric mean of a and b is the positive number x such
that , or x2 = ab.
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
4 and 25
Let x be the geometric mean.
x2 = (4)(25) = 100
x = 10
Find the geometric mean of each pair of numbers. If necessary, give the answer in simplest radical form.
Let x be the geometric mean.
x2 = (5)(30) = 150
5 and 30
Identify the Similar Triangles:
Z
W
∆UVW ~ ∆UWZ ~ ∆WVZ.
Identify the Similar Triangles:
EGF ～ EHG ～ GHF
A roof has a cross section that is a right triangle. The diagram shows the approximate dimensions of this cross section. Find the height h of the roof.
h = 23.1 feet
You can use Right Triangle Similarity Theorem to write proportions comparing the side lengths of the triangles formed by the altitude to the hypotenuse of a right triangle.
Find the missing length
Given: x = altitude, 16 and 9 are segments of the hypotenuse
x is the geometric mean (G.M.) of 9 and 16
x² = (9)(16)
x² = 144
x = 12
Find the missing length
Given: y is a leg, 9 and 2 are segment of the hyp
y is the G.M. of 11 (hyp) and 9 (adj leg)
y² = (11)(9)
y² = 99
y = 9.95
Find the missing length
Given: t is the hyp, 49 is a leg, 7 is a seg of the hyp.
49 is the G.M. of t and 7.
49² = 7t
2401 = 7t
t = 343
Find the missing length
Given: 6 is the altitude, 3 and a+4 are segments of the hyp.
6 is the G.M. of 3 and a+4
6² = 3(a+4)
36 = 3a + 12
24 = 3a
a = 8