9.3 Altitude-on-Hypotenuse Theorems

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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

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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.

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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.

Altitudes
4
Altitudes

• Notice that the third altitude creates two
  smaller right triangles. Is there something
  special about the three triangles?

Altitudes
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Index Card Activity

• Step 1 On your index card, draw a diagonal.
  Cut along this diagonal to separate your right
  triangles.

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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.

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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.

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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.

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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.

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Index Card Activity

• Step 3
• Label the angles of each triangle to match
  the diagram.
• Be sure to label all three triangles.
• In fact, use 4 colored markers to color edges and
  altitude, and label the back of each of them too.

B
A
C
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Index Card Activity

• Step 4 Arrange all three triangles so that they
  are nested. What does this demonstrate? Why
  must this be true?

C
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Index Card Activity

• Step 5 Write a similarity statement involving
  the large, medium, and small triangles.

C
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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.

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Example 1

• Identify the similar triangles in the diagram.

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Example 2

• Find the value of x.

Did you get x 12 ?
5
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Geometric Mean

• The geometric mean of two positive numbers a and
  b is the positive number x that satisfies
• This is just the square root of their product!

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Example 3

• Find the geometric mean of 12 and 27.

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Example 4

• Find the value of x.

Did you get x 18?
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Example 5

• The altitude to the hypotenuse divides the
  hypotenuse into two segments.
• What is the relationship between the altitude and
  these two segments?

altitude
Segment 1
Segment 2
hypotenuse
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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.

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Geometric Mean Theorem I
Heartbeat

• 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
x
b
a
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Example 6

• Find the value of w.

(w 9)(w 9) (8)(18)
w2 18w 81 144
(short leg) 8 w 9
(long
leg) w 9 18
w2 18w - 63 0
(w 21)(w 3) 0
w -21, 3
8 (3 9) (3 9) 18 8 12 12 18
2 3 2 3 v
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Example 7

• Find the value of x.

Boomerang!
(short leg) 3 x

(hypotenuse) x 12
x2 36
Did you get x 6?
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Geometric Mean Theorem II

• Geometric Mean (Leg) Theorem
• When an altitude is drawn to the hypotenuse of a
  right triangle, each leg is the geometric mean
  between the hypotenuse and the segment of the
  hypotenuse that is adjacent to the leg.

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Geometric Mean Theorem II
Boomerang

• Geometric Mean (Leg) Theorem
• When an altitude is drawn to the hypotenuse of a
  right triangle, each leg is the geometric mean
  between the hypotenuse and the segment of the
  hypotenuse that is adjacent to the leg.

a
a
x
c
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Geometric Mean Theorem II
Boomerang

• Geometric Mean (Leg) Theorem
• When an altitude is drawn to the hypotenuse of a
  right triangle, each leg is the geometric mean
  between the hypotenuse and the segment of the
  hypotenuse that is adjacent to the leg.

b
b
y
c
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Example 8

• Find the value of b.

Did you get b 25 ?
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Example 9

• Find the value of the variable.

w k
6
4
Boomerang!
HEARTBEAT
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9.3 Assignment

• P 379
• (1 5 8 13 16, 17)