The Question: A central angle of two concentric circles is $\frac{3\pi}{14}$. The area of the large sector is twice the area of the small sector. What is the ratio of the lengths of the radii of the two circles?
Answer: 0.71:1
The answer I end up with is $\sqrt{2}$:$1$, instead of $\frac{\sqrt{2}}{2}$: 1 (which I'm assuming is how they got 0.71). My book says the angle measure is superfluous, and that "areas of similar figures are proportional to the squares of linear measures associated with those figures". Using the similar figures property, this was my answer:
if $r_2$ = radius of larger circle and $r_1$ = radius of smaller circle and likewise for $a_2$ and $a_1$, then ($\frac{r_2}{r_1}$)$^2$ = $\frac{a_2}{a_1}$. Since $\frac{a_2}{a_1}$ = 2, taking the root of both sides yields $\sqrt{2}$ = $\frac{r_2}{r_1}$. Any explanation would be appreciated, thank you very much!
Best Answer
The question does not specify which radius comes first in the proportion. The book answer gives $r_1:r_2=\frac {\sqrt 2}2:1$ while you are saying $r_2:r_1=\sqrt 2:1$. You agree with the book but present the data in the opposite order. I also object strongly to saying $\frac {\sqrt 2}2=0.71$ but that is another issue.