Use the buttons in the app below to explore the construction of a golden rectangle.[br][br]When the construction is done, drag points [i]A[/i] and [i]B[/i] and observe how the ratio of the lengths of the rectangle sides is always equal to the golden number [math]\varphi=\frac{1+\sqrt{5}}{2}\approx1.61803[/math]
In the app below, there is a golden rectangle, constructed exactly as described in the previous app.[br]You already know that the ratio between side [math]AF[/math] and side [math]FG[/math] of the rectangle is constant and equal to [math]\varphi[/math](the golden ratio).[br]Imagine removing the square [math]ABCD[/math] from the rectangle, and consider the remaining rectangle [math]BFGC[/math].[br]Measure the lengths of its sides using GeoGebra tools, then calculate the ratio between the longer side and the shorter side.[br]What do you observe?
Describe the results that you have obtained.
Use GeoGebra's tools to build another golden rectangle, starting from the square with side [math]BF[/math].[br]When done, imagine to remove the square you have built, then measure the sides of the remaining rectangle and calculate their ratio.[br]What do you observe?
Subtracting from a golden rectangle a square with a side equal to the shorter side of the rectangle produces another golden rectangle.[br][br]And the ratio between the sides of two consecutive squares removed from the golden rectangle is golden!