A 12 inch line segment is divided into two parts. Which of the following lengths result in a ratio closest to the golden ratio, 1+√5 OVER 2?
A. 6 inches and 6 inches
B. 7 inches and 5 inches
C. 7.5 inches and 4.5 inches
D. 7.75 inches and 4.25 inches
We want to find which of the following values is closest to $\frac{1+\sqrt{5}}{2}$. Let $x$ be the length of the first segment, so the length of the second segment is $12-x$. Then we want the ratio $\frac{x}{12-x}$ to be closest to $\frac{1+\sqrt{5}}{2}$. We can simplify this ratio as follows: \begin{align*}
\frac{x}{12-x}&=\frac{1}{\frac{12}{x}-1}\\
&=\frac{1}{\frac{12-x}{x}}\\
&=\frac{x}{12-x}.
\end{align*} Setting $\frac{x}{12-x}=\frac{1+\sqrt{5}}{2}$ and simplifying gives $x=7+\sqrt{5}$ (note that we reject the solution $x=7-\sqrt{5}$ since $x$ must be greater than 6). Therefore, the two segments are approximately 7.38 inches and 4.62 inches. The answer choice that is closest to these values is $\boxed{\textbf{(D) }7.75\text{ inches and }4.25\text{ inches}}$.