Key Learning


Solve problems involving similar right
triangles formed by the altitude drawn to the
hypotenuse of a right triangle.
Use a geometric mean to solve problems such
as estimating a climbing distance.
Proportions in right triangles

In Lesson 8.4, you learned
that two triangles are
similar if two of their
corresponding angles are
congruent. For example P
∆PQR ~ ∆STU. Recall
that the corresponding
side lengths of similar
triangles are in
proportion.
S
U
R
T
Q

If the altitude is drawn
to the hypotenuse of a
right triangle, then the
two triangles formed
are similar to the
original triangle and to
A
each other.
C
D
∆CBD ~ ∆ABC, ∆ACD ~ ∆ABC, ∆CBD ~ ∆ACD
B
Finding the Height of a Roof



Roof Height. A roof has a
cross section that is a right
angle. The diagram shows
the approximate
dimensions of this cross
section.
A. Identify the similar
triangles.
B. Find the height h of the
roof.
Solution:

Y
You may find it helpful to
sketch the three similar
3.1 m
h
triangles so that the
corresponding angles and X
W
sides have the same
orientation. Mark the
congruent angles. Notice
that some sides appear in
5.5 m
more than one triangle.
For instance XY is the
hypotenuse in ∆XYW and
the shorter leg in ∆XZY.
Y
h
∆XYW ~ ∆YZW ~ ∆XZY.
Z
6.3 m
Z
X
W
3.1 m
5.5 m
Y
Solution for b.

Use the fact that ∆XYW ~ ∆XZY to write a
proportion.
YW
ZY
=
XY
XZ
Corresponding side lengths are in
proportion.
h
5.5
=
3.1
6.3
Substitute values.
6.3h = 5.5(3.1)
h ≈ 2.7
Cross Product property
Solve for unknown h.
The height of the roof is about 2.7 meters.
Using a geometric mean to solve
problems

In right ∆ABC,
altitude CD is drawn
to the hypotenuse,
forming two smaller
right triangles that
are similar to ∆ABC
From Theorem 9.1,
you know that ∆CBD
~ ∆ACD ~ ∆ABC.
C
A
D
B
B
C
D
C
D
A
A
B
C
C
A
D
B
B
C
D
C
A
D
B
Notice that CD is the longer leg
of ∆CBD and the shorter leg
of ∆ACD. When you write a
proportion comparing the
legs lengths of ∆CBD and
∆ACD, you can see that CD
is the geometric mean of BD
and AD.
Longer leg of ∆CBD.
Shorter leg of ∆CBD.
BD
CD
A
C
Shorter leg of ∆ACD
=
CD
AD
Longer leg of ∆ACD.
C
A
D
B
B
C
D
C
A
Sides CB and AC also appear in
more than one triangle. Their
side lengths are also geometric
means, as shown by the
proportions below:
D
B
BD
CD
Hypotenuse of ∆CBD
A
Shorter leg of ∆ABC.
Hypotenuse of ∆ABC.
C
=
CD
AD
Shorter leg of ∆CBD.
C
A
D
B
B
C
D
C
A
Sides CB and AC also appear in
more than one triangle. Their
side lengths are also geometric
means, as shown by the
proportions below:
D
B
BD
CD
Hypotenuse of ∆ACD
A
Longer leg of ∆ABC.
Hypotenuse of ∆ABC.
C
=
CD
AD
Longer leg of ∆ACD.
Geometric Mean Theorems


In a right triangle, the altitude from the
right angle to the hypotenuse divides
the hypotenuse into two segments. The
length of the altitude is the geometric
mean of the lengths of the two
segments
In a right triangle, the altitude from the
right angle to the hypotenuse divides
the hypotenuse into two segments. The
length of each leg of the right triangle is
the geometric mean of the lengths of
the hypotenuse and the segment of the
hypotenuse that is adjacent to the leg.
C
A
D
BD
=
CD
CD
AD
AB
CB
=
CB
DB
AB
AC
=
AC
AD
B
What does that mean?
2
x
6
6
=
x
18 = x2
√18 = x
5
y
3
x
5+2
3
y
7
y
=
y
=
2
y
2
√9 ∙ √2 = x
14 = y2
3 √2 = x
√14 = y