Thales

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Thales Theorem
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Thales Theorem - it is the most important
theoremin Euclidean geometry.

• Thales of Miletus- ( c.625 c.547 BC)
• Greek mathematician and philosopher who is
  generally considered to
• be the first Western scientist and
  philosopher.His fame as a
• mathematician rests upon his suppose discovery of
  seven geometrical
• propositions, including the familiar Euclidean
  theorems. According
• to one tradition, Thales acquired his
  mathematical learning from
• Egyptian scholars. He is reported to have
  predicted the solar eclipse of
• 585 BC
• More about Thales you can find on our website
  www.mathematics.xt.pl
• ( presentation about Greek mathematicians)

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Thales TheoremIf arms og an angle are cut by
parallel straight lines, then the ratio of the
lengths of the line segments obtained on one arm
are equal to the corresponding segments obtained
on the second arm.

• For example
• OAOB O'A'O'B'
• OAAB O'A'A'B'
• But also
• OCAC O'C'A'C'
• Conclusion
• In the situation like on the drawing , from
  Thales Theaorem it follows that OAOB
  AA'BB'

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Converse of a given theorem.Is obtained by
switching, the premise and the proposition. As
the converse does not necessarily hold, all the
more reason to consider the cases it does.

• Converse of Thales Theorem
• States that if arms of an angle are cut by
  several straight lines and the ratios of the
  lenghts of the line segments obtained on one arm
  are equal to the corresponding segments obtained
  on rhe second arm, the those straight lines are
  parallel.

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ApplicationsThalesTheorem has found a number of
applications. Let us just a few

• Pyramids height
• The legend has it than Thales amazed Egyptian
  priests by calculating the height of the Great
  pyramid using a stick and its shadow. This is
  how he did it.
• From ThalesTheorem we have the proportion
• OAOB AA'BB' so BB'AA'OBOA.
  
• Knowing AA' the sicks lenght, and measuring
  OA its shadows OB length, we can obtain
  the pyramids height of any large object.

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• Ship-to-shore distance Using a bit different
  method we can calculate the offshore disance of a
  ship.
• Using ThalesTheorem we have
• (A'Ax)B'A' xBA skad xA'ABA(B'A'
  -BA).
• Be measuring lengths of all segments in the above
  equation we get affshore distance x

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• Division of a sehment in a given ratio.
• Let us be given two segments of length a and b.
  Our task is to divide a given segment AB in the
  ratio a b.
• Fram the picture, using ThalesTheorem we see
  that the point P divides the segment AB in the
  given ratio. The above construct was fundamental
  in mathematics of Ancient Greece. It facilitated
  multiplication and diision of segments which the
  Greeks identified with numbers.

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Similarity of Trangles  Are conditions (both
sufficient and necessary) for two trangles to be
congruent. There are several criteria of
conruence

• I Feature of similarityof triangles
• Two triangles are congruent if they share two
  corresponding angles

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II Feature of similarity of triangles

• Two tringles are congruent if their corresponding
  side are equa.

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III Feature of similarity of triangles

• Two triangles are congruent if a pair of
  orresponding side and the included angle are
  equal.

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Features of similarity of triangles
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Tasks

• 1. Let two straight lines AC and BD be parallel
• a) OA4cm,OC3cm,AB1,6cm, oblicz CD
• b) OD4,8cm, OA2cm, AB4cm, oblicz OC
• 2. Calculate the height of the Tower of the Wends
  (Athens) if its shadow is 10 m long, and at the
  same time a stick 2.6 m londg casts a two-meter
  shadow.

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• 3. Calculate the height of a tree if its shadow
  is 12 m long and its crowns shadow is 8 m long.
  The lowest branches are at 2 m.
• 4. Let us be given a triangle ABC.A straight line
  parallel to the AB side cuts the AC side passing
  through tje point M and cut the BC side passing
  throught the point N. Calculate the length BN and
  NC, if
• AM/MC2/3 i BC 10 cm.

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• 5. If straight lines AB, CD, EF are parallel to
  one another, calculate
• OB , if OC7cm, OA3cm, BD2cm
  
• OE , if AF9cm, OA3cm, OB5cm
• DB , ifi AC2cm, OC3cm, OB5cm