Answer
a. $\sqrt{5} $
b. $\sqrt{5} $
c. $1+\sqrt{5} $
d. $1.61803398875$
Work Step by Step
Let $a$ denote the length of the sides of the square $ABCD$
$\therefore AB = BD = CD = AC = a = 2$
From the rectangle $ACEF:\,AC = EF = a$
Let $b$ denote the length of $DE$
The golden rectangle $ACEF$ must satisfy the relationship:
$$\frac{a+b}{a} = \frac{a}{b}$$
a.
$OB$ can be found from the triangle $ODB$
$$OB = \sqrt{(OD)^2+(DB)^2}$$
$$OB = \sqrt{(1)^2+(2)^2}=\sqrt{5}$$
b.
To find the length of $OE$, we need to find the length of $DE$ ($b$) which can be obtained from the golden ratio equation.
$$b^2+ab-a^2=0$$
$$b^2+2b-4=0$$
$b$ can be obtained by using the quadratic formula
$$\frac{-b\pm\sqrt{b^2-4ac}}{2a}$$
$$\frac{-2\pm\sqrt{2^2-4\times1\times-4}}{2\times1} = \frac{-2\pm\sqrt{20}}{2}=\frac{-2\pm2\sqrt{5}}{2} $$
$$\therefore b = -1+\sqrt{5}$$
The other solution is rejected as it would yield a negative length.
$$OE = OD +DE = \frac{a}{2}+b = 1 -1 +\sqrt{5} = \sqrt{5} $$
c.
$$CE = CD+DE = a+b = 1+\sqrt{5}$$
d.
$$\frac{CE}{CF} = \frac{1+\sqrt{5}}{2} = 1.61803398875 \, (Golden \, Ratio)$$
