Transcript Geometric Mean Theorems

Similar Right Triangles
Chapter 8
Activity: Investigating similar right
triangles. Do in pairs or threes
1.
2.
3.
Cut an index card along one of
its diagonals.
On one of the right triangles,
draw an altitude from the right
angle to the hypotenuse. Cut
along the altitude to form two
right triangles.
You should now have three
right triangles. Compare the
triangles. What special
property do they share?
Explain.
Theorem 8.1 (page 518)

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
Proportions in right triangles

In chapter 7, 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
A
D
C
B
D
D
C
A
They ALL look the same!
B
C
In right ∆ABC, altitude
CD is drawn to the
hypotenuse, forming
two smaller right
triangles that are
similar to ∆ABC From A
Theorem 8.1, you know
that
∆CBD~∆ACD~∆ABC.
D

C
B
Using a geometric mean to solve
problems
Similar = same shape, different size
Geometric Mean Theorems


Theorem 8.2: 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
Theorem 8.3: 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.
Geometric Mean Theorems
C
If you want to find
the Altitude… use
Geometric Mean
BD
GM
A
D
CD is the Geometric Mean
of AD
and BD
CD
B
=
CD
AD
Example: Use Geometric Mean to find
the Altitude of the Triangle
x
6
6
=
x
18 = x2
√18 = x
√9 ∙ √2 = x
3 √2 = x
3
x
3
Geometric Mean Theorems
C
If you want to find the side on the
right side of the triangle… use
Geometric Mean
GM
A
A
D
CB is the Geometric Mean
of DB
and AB
B
B
AB
CB
=
CB
DB
Example: Find y (the right leg of the
triangle) using Geometric Mean
2
y
5
5
5+2
=
y
7
=
y
14 = y2
√14 = y
y
2
y
2
2
y
Y is the leg on
the right side
Rotate the triangle to help
visualize what part you need
Geometric Mean Theorems
C
If you want to find the side on the
LEFT side of the triangle… use
Geometric Mean
GM
A
A
D
AC is the Geometric Mean
of AD
and AB
B
B
AB
AC
=
AC
AD
Example: Find y (the left leg of the
triangle) using Geometric Mean
35
35
y
=
245 = y2
√245 = y
7√5 = y
y
7
7
y
Y is the leg on
the Left side
Ex. 1: 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.