We want to compare the ratios $\frac{a}{b}$ and $\frac{a+b}{a}$ to $\frac{1+\sqrt{5}}{2}$. Simplifying the golden ratio gives $\frac{2+\sqrt{5}}{2} = 1 + \frac{\sqrt{5}}{2}$. We want the difference between our ratio and the golden ratio to be as small as possible, so we want $\left| \frac{a}{b} - \frac{2+\sqrt{5}}{2}\right|$ and $\left| \frac{a+b}{a} - \frac{2+\sqrt{5}}{2}\right|$ to be small.
For the answer choices:
A. $\frac{6}{6} = 1$, which is not close to the golden ratio.
B. $\frac{7}{5} = 1.4$, and $\frac{12}{7} = 1.\overline{714}$, so choice B is not a good candidate either.
C. $\frac{7.5}{4.5} = 1.\overline{6}$, which is the best so far, and $\frac{12}{7.5} = 1.6$, which is also close.
D. $\frac{7.75}{4.25} = 1.\overline{8235}$ is farther from the golden ratio than option C.
Therefore, the answer is $\boxed{\textbf{(C)}\ 7.5 \text{ inches and 4.5 inches}}$.
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
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
1 answer
1 answer