7-3: Similar Right Triangles
Objectives:
1. To find missing
lengths in similar
right triangles
Assignment:
• P. 361: 32, 34, 36
• P. 453-456: 1-3, 5-23,
30, 31, 33, 38, 39
• Challenge Problems
Objective 1
You will be able to find missing
lengths in similar right triangles
Altitudes
Recall that an altitude
is a segment drawn
from a vertex that is
perpendicular to the
opposite of a
triangle. Every
triangle has three
altitudes.
Altitudes
In a right triangle, two of these altitudes are
the two legs of the triangle. The other one
is drawn perpendicular to the hypotenuse.
B
AB
Altitudes:
BC
BD
A
D
C
Altitudes
Notice that this third altitude creates three
right triangles. Is there something special
about those triangles?
B
AB
Altitudes:
BC
BD
A
D
C
Index Card Activity
Step 1: On your index card, draw a
diagonal. Cut along this diagonal to
separate your right triangles.
Index Card Activity
Step 2: On one of
the right triangles,
fold the paper to
create the altitude
from the right angle
to the hypotenuse.
Cut along this fold
line to separate
your right triangles.
Index Card Activity
Step 2: On one of
the right triangles,
fold the paper to
create the altitude
from the right angle
to the hypotenuse.
Cut along this fold
line to separate
your right triangles.
Index Card Activity
Step 2: On one of
the right triangles,
fold the paper to
create the altitude
from the right angle
to the hypotenuse.
Cut along this fold
line to separate
your right triangles.
Index Card Activity
Step 3: Notice that
the small and
medium triangle
can be stacked on
the large triangle
so that the side
they share is the
third altitude of the
large triangle.
Index Card Activity
Step 3: Label the
angles of each
triangle to match
the diagram. Be
sure to label all
three triangles. In
fact, label the
back of each of
them too.
B B
A
D D
C
Index Card Activity
Step 4: Arrange all
three triangles so
that they are
nested. What
does this
demonstrate?
Why must this be
true?
A
B
A
B
D
D
B
Index Card Activity
Step 5: Write a
similarity
statement
involving the
large, medium,
and small
triangles.
A
B
A
B
D
D
B
ABC ~ BDC ~ ADB
Right Triangle Similarity Theorem
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 each other.
Example 1
Identify the similar
triangles in the
diagram.
Example 2
Find the value of x.
Geometric Mean
The geometric mean of two positive numbers
a and b is the positive number x that
satisfies
a x

x b
2
So x  ab
And x  ab
This is just the square root of their product!
Example 3
Find the geometric mean of 12 and 27.
Example 4
Find the value of x.
x
12
27
Example 5
The altitude to the
hypotenuse divides
the hypotenuse into
two segments.
What is the
relationship
between the altitude
and these two
segments?
x
12
27
Geometric Mean Theorem I
Geometric Mean (Altitude)
Theorem
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.
x
a
b
a x

x b
Geometric Mean Theorem I
Geometric Mean (Altitude)
Theorem
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.
Heartbeat
xx
aa
bb
a x

x b
Example 6
Find the value of w.
Example 7
Find the value of x.
3
x
12
Geometric Mean Theorem II
Geometric Mean (Leg)
Theorem
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

a x
a
x
c
b
y
c
c b

b y
Geometric Mean Theorem II
Geometric Mean (Leg)
Theorem
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.
Boomerang
c a

a x
aa
xx
cc
b
y
c
c b

b y
Geometric Mean Theorem II
Geometric Mean (Leg)
Theorem
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.
Boomerang
c a

a x
a
x
c
bb
yy
cc
c b

b y
Example 8
Find the value of b.
Example 9
Find the value of variable.
1. w =
2. k =
7-3: Similar Right Triangles
Objectives:
1. To find missing
lengths in similar
right triangles
Assignment:
• P. 361: 32, 34, 36
• P. 453-456: 1-3, 5-23,
30, 31, 33, 38, 39
• Challenge Problems