The Golden Ratio

Problem: Find point P on segment AB such that AP:PBAB:AP.


If we let r be the length of AP and s be the length of PB, then the ratio above can be written as

\displaystyle\frac{r}{s} =\frac{r+s}{r}

Simplifying the equation we have r^2 = rs + s^2 which is equivalent to r^2-rs - s^2, a quadratic equation in r. So, we solve for r using the quadratic formula with a = 1, b = s and c = -s^2:

r = \displaystyle\frac{-b \pm \sqrt{b^2-4ac}}{2a} = \frac{-s \pm \sqrt{s^2-4(1)(-s^2)}}{2(10)}.

Simplifying this gives us r = \frac{s + s\sqrt{5}}{2}. Dividing both sides by s, we have

\displaystyle\frac{r}{s} = \frac{1 + \sqrt{5}}{2} \approx 1.61803398874989. » Read more