(2) By Pythagoras Theorem: MC²=125 so MC = 5Ö5, AE = 5 + 5Ö5 and BE = 5Ö5 - 5. Hence

AE AD = 5 + 5Ö5 10 = 1 + Ö5 2

and

BC BE = 10 5Ö5 - 5 = 2 Ö5 - 1 .

In order to write the last expression with a whole number in the denominator we multiply top and bottom by Ö5 + 1 which gives

BC BE = 2 Ö5 - 1 = 2(Ö5 + 1) (Ö5 - 1)(Ö5 + 1) = Ö5 + 1 2 .

Hence the two ratios are indeed equal.

(3) As the sides of the square are 1 unit, that is BC=1 unit, and BC/BE=f, then BE=1/f and AE = 1 + 1/f. We have also AE/AD = f so

f = 1 + 1 f .

Substituting f = 1 the left hand side of this expression is less than the right hand side. If we increase the value given to f the left hand side increases and the right hand side decreases continuously. Substituting f = 2 the left hand side is greater than the right hand side so the value of f which satisfies this equation must lie between 1 and 2.

(4)Reading from the graph the values are approx 1.62 and -0.62

(5)Multiplying through by f this equation becomes:

f2 - f- 1 = 0

which has solutions f = (1±Ö5)/2 and taking the positive value this gives the golden ratio f = (1 + Ö5)/2.