# Geo 9 3 Altitude on Hypotenuse Thm ppt comp

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```					9.3 Altitude-on-Hypotenuse
Theorems
Objectives:
1. To find the geometric mean of two
numbers.
2. To find missing lengths of similar right
triangles that result when an altitude is
drawn to the hypotenuse
Altitudes
Recall that an altitude
is a segment drawn
from a vertex such
that it is
perpendicular to the
opposite side of the
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 the third altitude creates two
smaller right triangles. Is there something
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
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
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
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
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:
B
Label the angles of each
triangle to match the
diagram.                   A         C
Be sure to label all three
triangles.
In fact, use 4 colored            B B
markers to color edges
and altitude, and label
the back of each of them
too.                       A   D D       C
Index Card Activity
Step 4: Arrange all
three triangles so
that they are        A
B
nested. What
does this                    A

demonstrate?
B   D   D   B   C
Why must this be
true?
Index Card Activity
Step 5: Write a
similarity
statement           A
B
involving the
large, medium,              A

and small
B   D   D     B   C
triangles.
Right Triangle Similarity Theorem

If an 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.

Did you get x = 12   ?
5
Geometric Mean
The geometric mean of two positive numbers
a and b is the positive number x that
satisfies        a x

x b
So x  ab
2

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

Did you get x = 18?
Example 5
The altitude to the
hypotenuse divides                     altitude
the hypotenuse into
two segments.
x
What is the
relationship           Segment 1
12
Segment 2
27
between the altitude          hypotenuse
and these two
segments?
Geometric Mean Theorem I
Geometric Mean (Altitude)
Theorem
In a right triangle, the altitude       x
from the right angle to the
hypotenuse divides the            a        b

hypotenuse into two
segments.                                 a x
The length of the altitude is the            
x b
geometric mean of the
lengths of the two segments.
Geometric Mean Theorem I
Geometric Mean (Altitude)
Heartbeat
Theorem
In a right triangle, the altitude       x
x
from the right angle to the
hypotenuse divides the            a
a        b
b

hypotenuse into two
segments.                                 a x
The length of the altitude is the            
x b
geometric mean of the
lengths of the two segments.
Example 6
Find the value of w.            (w + 9)(w + 9) = (8)(18)

w2 + 18w + 81 = 144
(short leg)   8         w+9
=            w2 + 18w - 63 = 0
(long leg)    w+9       18
(w + 21)(w – 3) = 0

w = {-21, 3}

8 : (3 + 9) = (3 + 9) : 18
8 : 12 = 12 : 18
2:3 = 2:3√
Example 7
Find the value of x.        Boomerang!
(short leg)      3        x
=
(hypotenuse)     x        12

3      12
x2 = 36
x

Did you get x = 6?
Geometric Mean Theorem II
Geometric Mean (Leg)                                c a
Theorem                    a

a x
When an altitude is drawn
x
to the hypotenuse of a
c
right triangle, each leg
is the geometric mean
c b
between the hypotenuse                         b

and the segment of the                             b y
hypotenuse that is                         y

Geometric Mean Theorem II
Geometric Mean (Leg)
Boomerang
Theorem                  a
a
c a

a x
When an altitude is drawn
to the hypotenuse of a     x
x

right triangle, each leg           c
c

is the geometric mean
c b
between the hypotenuse                     b

and the segment of the                         b y
hypotenuse that is                     y

Geometric Mean Theorem II
Geometric Mean (Leg)
Boomerang
Theorem                    a
c a

a x
When an altitude is drawn
x
to the hypotenuse of a
c
right triangle, each leg
is the geometric mean
c b
between the hypotenuse                         b
b

and the segment of the                             b y
hypotenuse that is                         y
y

c
Example 8
Find the value of b.

Did you get b = 25   ?
6
Example 9
Find the value of the variable.
1. w = 6                2. k = 4

HEARTBEAT               Boomerang!
9.3 Assignment

P 379:
(1 – 5; 8 – 13; 16, 17)

```
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