Geometric mean

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Geometric mean

• When an altitude is drawn from the vertex of a
  right triangle's 90 degree angle to its
  hypotenuse, it splits the triangle into 2 right
  triangles that exhibit a special relationship.

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Theorem 50-1

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the 2 triangles formed are
  similar to the original triangle

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CAUTION !!!

• Theorem 50-1 is only true if the altitude of the
  right triangle has an endpoint on the hypotenuse,
  not on the triangle's legs

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Identifying similar right triangles

• Find RS and RQ,- Tri PQR is similar to Tri PSQ is
  similar to Tri RSQ

P
S
3
5
4
R
Q
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Finding geometric mean

• Sometimes , the means of a proportion are equal
  to one another.
• This is a special kind of proportion that can be
  used to find the geometric mean of 2 numbers
• The geometric mean for positive numbers a and b,
  is the positive number x , such that

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Another way to state geometric mean

• The geometric mean of a and b is equal to the
  square root of the product of a and b, since
• ab x2

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Find the geometric mean

• Find the geometric mean of 3 and 12
• Find the geometric mean of 4 and 16
• Find the geometric mean of 2 and 9 in simplified
  radical form.
• Find the geometric mean of 5 and 11 to nearest
  tenth.

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Corollary 50-1-1

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the length of the altitude
  is the geometric mean between the segments of the
  hypotenuse

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Corollary 50-1-2

• If the altitude is drawn to the hypotenuse of a
  right triangle, then the length of a leg is the
  geometric mean between the hypotenuse and the
  segment of the hypotenuse that is closer to that
  leg.
• AZ YZ
• YZ XZ

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Find missing values a and ba 3 b43 5
4 5
4
3
a
b
5
11
Find missing value for y

• 3 y
• y 4/3

y
4/3
3
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