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
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 $\boxed{\textbf{(C)}\ 7.5 \text{ inches and 4.5 inches}}$.
To find the ratio closest to the golden ratio, we need to calculate the ratio for each option and compare them.
The golden ratio, often denoted by the Greek letter phi (φ), is approximately 1.6180339887.
Let's calculate the ratio for each option:
A. 6 inches and 6 inches:
Ratio = 6/6 = 1
B. 7 inches and 5 inches:
Ratio = 7/5 = 1.4
C. 7.5 inches and 4.5 inches:
Ratio = 7.5/4.5 ≈ 1.6666666667
D. 7.75 inches and 4.25 inches:
Ratio = 7.75/4.25 ≈ 1.8235294118
The ratio that is closest to the golden ratio is option D (7.75 inches and 4.25 inches), with a ratio of approximately 1.8235294118.
To determine which lengths result in a ratio closest to the golden ratio, we need to calculate the ratios for each option. The golden ratio, also known as phi (∅), is approximately equal to 1.618.
To calculate the ratio, we divide the longer segment by the shorter segment.
A. 6 inches and 6 inches: The ratio is 6/6 = 1.
B. 7 inches and 5 inches: The ratio is 7/5 = 1.4.
C. 7.5 inches and 4.5 inches: The ratio is 7.5/4.5 = 1.666.
D. 7.75 inches and 4.25 inches: The ratio is 7.75/4.25 ≈ 1.8247.
Now, we compare the calculated ratios to the golden ratio (∅ ≈ 1.618) to determine which is closest:
A. The ratio is 1, which is less than the golden ratio.
B. The ratio is 1.4, which is greater than the golden ratio.
C. The ratio is 1.666, which is greater than the golden ratio.
D. The ratio is approximately 1.8247, which is also greater than the golden ratio.
Therefore, none of the given options result in a ratio closest to the golden ratio.