Question

We can relate the golden ratio to the lengths of line segments as follows. If we start with a line segment AB and locate point C on the segment such that....

Answer 1

For part \(a\), start with the proportion \(\dfrac{AB}{AC}=\dfrac{AC}{CB}\)



replacing \(AB\) by \(AC+CB\), the proportion becomes
\[\begin{align}\dfrac{AC+CB}{AC}&=\dfrac{AC}{CB}\\~\\
1+\dfrac{CB}{AC}&=\dfrac{AC}{CB}\\~\\
\dfrac{CB}{AC}&=\dfrac{AC-CB}{CB}\\~\\
\dfrac{ AC}{CB}&=\dfrac{CB}{AC-CB}
\end{align}\]