THALES' THEOREM
Geometry
 
1. GEOMETRIC RATIO

Four segments a, b, c and d are in proportion when the following relation is true: a/b=c/d. This equal quotient is referred to as proportion or ratio. The Descartes window draws the segments written on the line underneath and indicates the ratios for each pair of values.
1.- Three lines whose lengths are a=5cm, b=7cm, c=10cm and line d, whose length is unknown, are in proportion. Work out the ratio and the unknown length of line d.
 

Write the known values directly into the first three boxes and use the arrows to find the unknown value. Remember that the ratios are always the same.

2.- Repeat the exercise for the following values: 5, 4, 3 and d.

When the two middle terms are equal, i.e. when b=c, this repeated value is known as the mean proportion or geometric mean. In order to find the mean proportion when two values a and d are given, you need to find this repeated value.
3.- Use the Descartes window to work out the mean proportion of the values 2 and 8. Work out the mean proportion of 4 and 6 too. Do you know another way you can use to work out this value?
2.THALES' THEOREM
If several parallel straight lines are cut by two transversal lines, the ratio of any two segments of one of these transversals is equal to the ratio of the corresponding segments of the other transversal. The following Descartes window shows how three parallel straight lines are cut by two secants r and s. The window indicates the length of the segments in these two transversals at any time, and that the ratio of the segments does not change.
4.- Look carefully at how AB/BC=A'B'/B'C' is always true. Move points A, A', C and C' and see if the ratio values change.

5.- Move the parallel line in the middle by dragging the red point with the mouse and notice how each of the quotient values change.

6.- Copy this example into your exercise book, with the same measurements, and measure the segments to see if the ratio values are equal or not.

7.- Move the parallel line in the middle until segment AB is equal to BC. Then, see if the segment A'B' is equal to B'C'. Move the lines r and s and see if the segments remain equal to each other.

3. A RESULT OF THALES' THEOREM
If points A and A' had met in the window above they would have formed a triangle with points C and C'. Thales' theorem would have still been satisfied which allows us to conclude that: Any line parallel to one of the sides of a triangle, which cuts the other two sides, produces segments which are in proportion to each other. In the Descartes' window there is a triangle ABC and a parallel line to side BC which passes through points D and E producing segments which are in proportion to each other, as they are in the same ratio.

8.- Click on the Init button and draw an identical triangle to the one in the window into your exercise book. Draw the line parallel to side BC and check the measurements and ratio values. Verify that the ratio values stay the same when points B and C are moved horizontally, but that they do change when the we move the straight line.

9.- Move the straight line onto vertex A and note that the segments are still in proportion to each other. This is also the case when the line is dragged below points B and C.

10.- A triangle has sides AB=10 cm, AC=12 cm and BC=8 cm. A line parallel to side BC is drawn 4 cm away from vertex A along side AB, which cuts the triangle at points D and E. Work out the lengths of AD, AE and DE.
       
           
  Miguel García Reyes
 
Spanish Ministry of Education. Year 2001
 
 

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