«A straight line is divided into extreme and mean ratio when the whole line is to the larger segment what the larger segment is to the smaller one» The Elements, book II, proposition 11, Euclides. 

Given the segment PQ, we can say that PR is the segment or golden section of PQ when:

Write the values of PR and RQ in your exercise book. What is the ratio of proportion?

The number you have is the ratio of proportion and is called the "golden number". It can be represented by the Greek letter F.

(Click on the link to find out more about F ).

In order to work out the golden section of a segment PQ, we draw the perpendicular to it from Q and indicate the segment AQ, 1/2 the length of PQ, on this line. We join A and P together and find that AB = AQ. We draw an arc BR from the circle with centre P; R is the point which divides segment PQ into the golden mean.

3.- Work out the golden section of other segments, e.g. segments whose lengths are the following: 8, 10, 12 and 20. Write the results in your exercise book and check that the following ratio is always true:

The Greeks called a rectangle which is made up of a square ABCD and a rectangle CEPD, where rectangles ABEP and CEPD are similar, a golden rectangle. The sides of a golden rectangle are in golden ratio, i.e. their ratio (quotient) is the golden number.

4.- Construct a golden rectangle whose base is 18. Write the value of the height in your exercise book and check that your answer is the same as the one you got in the window.

5.- Construct more golden rectangles e.g. whose bases are 12 and 20. Write down the values of the height in your exercise book and check that your answers are the same as those given in the window above.

6.- Construct some more golden rectangles given their height e.g. 8 y 10. Write down the length of the base in your exercise book. Is the base the same length as that given in the window above? What is the same? Can you explain?

7.- Measure an ID card/bank card which is the same size as a credit card. Is it a golden rectangle?