Download Theorem 7-3

Study Guide 7 – 4
The geometric mean of two positive numbers a and b is a number x such that
a x
 .
x b
x  ab
Example 1: Find the geometric mean of 4 and 18.
Example 2: Find the geometric mean of
7 2 and 5 6 .
Example 3: 6 is the geometric mean between 4 and what value?
Draw the altitude CD of the right triangle below to create two additional right triangles.
A
C
C
C
S2
C
7
5
3
1
B
A
B
D
A
8
6
4
D
9
D
(a) Which angles have the same measure as  1?
(b) Which angles have the same measure as  2?
(c) Which angles have the same measure as  3?
(d) Based on the results, what is true about the three triangles?
(e) Complete the similarity statement. ∆ACB ~ ∆_________ ~ ∆_________
C
D
2
6
X
A
R
1
8
D
3
A
B
4
Y
5
C
Z
C
7
9
B
Because the triangles are similar, the corresponding sides are proportional. Here are three of those proportions:
AC
AB
AD CD
BC
AB



Between X and Y:
,
between Y and Z:
,
between X and Z:
AD AC
CD BD
BD BC
Theorem 7-3
The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are
______________ to the original triangle and _____________ to each other.
Corollary 1 to Theorem 7-3
The length of the altitude to the hypotenuse of a right triangle is the _____________________ of the lengths
of the segments of the hypotenuse.
Corollary 2 to Theorem 7-3
The altitude to the hypotenuse of a right triangle separates the hypotenuse so that the length of each leg of the
triangle is the geometric mean of the length of the _________________________ and the length of the
__________________.
If we are only given pictures, how are the sides of the triangles related to find the geometric mean?
B
Example 4: Solve for x.
Example 5: Solve for x and y.
Example 6: Solve for x.
y
12
16
y
6
x
x
x
15
5
2
20
Example 7: At a golf course, Maria drove her ball 192 yd straight toward
the cup. Her brother Gabriel drove his ball straight 240 yd, but not toward
the cup. The diagram shows the results. Find x and y, the remaining
distances from the cup.
y
240 yd
Example 8:
Complete each proportion us the triangle below:
x
192 yd
i)
r h

h ?
h is the geometric mean of what?
ii)
c a

a ?
a is the geometric mean of what?
iii)
s ?

? c
What is the geometric mean of s
a
b
h
r
s
c
and c?