1. Thales’ Theorem

2. Thales’ Theorem - it is the most important theorem in 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 www.mathematics.xt.pl ( presentation about Greek mathematicians)

3. Thales’ Theorem If 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: |OA|:|OB| = |O′A′|:|O′B′| |OA|:|AB| = |O′A′|:|A′B′| But also |OC|:|AC| = |O′C′|:|A′C′| Conclusion In the situation like on the drawing, from Thales’ Theaorem it follows that |OA|:|OB| = |AA′|:|BB′|

4. 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.

5. Applications Thales’Theorem has found a number of applications. Let us just a few: Pyramid’s height The legend has it than Thales amazed Egyptian priests by calculating the height of the Great pyramid using a stick and it’s shadow. This is how he did it. From Thales’Theorem we have the proportion: |OA|:|OB| = |AA′|:|BB′| so |BB′|=|AA′|·|OB|:|OA|. Knowing |AA′| –the sick’s lenght, and measuring |OA| – it;s shadow’s |OB| length, we can obtain the pyramid’s height of any large object.

6. Ship-to-shore distance Using a bit different method we can calculate the offshore disance of a ship. Using Thales’Theorem we have: (|A′A|+x):|B′A′| = x:|BA| skąd x=|A′A|·|BA|:(|B′A′|-|BA|). Be measuring lengths of all segments in the above equation we get affshore distance x

7. 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 Thales’Theorem 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.

8. 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

9. II Feature of similarity of triangles Two tringles are congruent if their corresponding side are equa.

10. III Feature of similarity of triangles Two triangles are congruent if a pair of orresponding side and the included angle are equal.

11. Features of similarity of triangles

12. Tasks 1. Let two straight lines AC and BD be parallel: a) [OA]=4cm,[OC]=3cm,[AB]=1,6cm, oblicz [CD] b) [OD]=4,8cm, [OA]=2cm, [AB]=4cm, oblicz [OC] 2. Calculate the height of the Tower of the Wends (Athens) if it’s shadow is 10 m long, and at the same time a stick 2.6 m londg casts a two-meter shadow. O C A D B

13. 3. Calculate the height of a tree if it’s shadow is 12 m long and it’s crown’s 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 cut’s 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/MC]=2/3 i [BC] = 10 cm.

14. 5. If straight lines AB, CD, EF are parallel to one another, calculate: a)[OB], if: [OC]=7cm, [OA]=3cm, [BD]=2cm b)[OE], if: [AF]=9cm, [OA]=3cm, [OB]=5cm c)[DB], ifi: [AC]=2cm, [OC]=3cm, [OB]=5cm E F O A B D