SIMILARITIES IN A RIGHT TRIANGLE

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SIMILARITIES IN A RIGHT TRIANGLE

• By SAMUEL M. GIER

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How much do you know
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DRILL

• SIMPLIFY THE FOLLOWING EXPRESSION.
• 1. 4.
• 
  
• 2.
• 5.
• 3.

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DRILL

• Find the geometric mean between the two given
  numbers.
• 1. 6 and 8
• 
  
• 2. 9 and 4

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DRILL

• Find the geometric mean between the two given
  numbers.
• 6 and 8
• h
• h 4
• 
  

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DRILL

• Find the geometric mean between the two given
  numbers.
• 2. 9 and 4
• h
• h 6
• 
  

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REVIEW ABOUT RIGHT TRIANGLES
A
LEGS

The perpendicular side
HYPOTENUSE
C
B
The side opposite the right angle
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SIMILARITIES IN A RIGHT TRIANGLE

• By SAMUEL M. GIER

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CONSIDER THIS
State the means and the extremes in the following
statement. 37 614  

  The means are 7 and 6 and the
extremes are 3 and 14.
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CONSIDER THIS
State the means and the extremes in the following
statement. 53 610  

  The means are 3 and 6 and the
extremes are 5 and 10.
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CONSIDER THIS
State the means and the extremes in the following
statement. ah hb  

  The means are h and the
extremes are a and b.
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CONSIDER THIS
Find h. ah hb  

applying the law of proportion. h² ab h
h is the geometric mean between a b.  

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THEOREMSIMILARITIES IN A RIGHT TRIANGLE

• States that In a right triangle, the altitude to
  the hypotenuse separates the triangle into two
  triangles each similar to the given triangle and
  to each other.

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ILLUSTRATION
?MOR ?MSO, ?MOR ?OSR by AA Similarity
postulate)

• In a right triangle (?MOR), the altitude to the
  hypotenuse(OS) separates the triangle into two
  triangles(?MOS ?SOR )each similar to the given
  triangle (?MOR) and to each other.

?MSO ?OSR by transitivity
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TRY THIS OUT!

• NAME ALL SIMILAR TRIANGLES

?ACD ?ABC ?ACD ?CBD ?ABC ?CBD
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COROLLARY 1.

• In a right triangle, the altitude to the
  hypotenuse is the geometric
• mean of the segments into which it divides
  the hypotenuse

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ILLUSTRATION

• CB is the geometric mean between AB BD.

In the figure,
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COROLLARY 2.

• In a right triangle, either leg is the geometric
  mean between the hypotenuse and the segment of
  the hypotenuse adjacent to it.

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ILLUSTRATION

• CB is the geometric mean between AB BD.

In the figure,