Given a square with area $A$. A circle lies inside the square, such that the circle touches all sides of the square. Another square with area $B$ lies inside the circle, such that all its vertices lie on the circle. Find the value of $\frac{A}{B}.$
Problem
Source:
Tags: geometry, circles, square
19.09.2020 03:58
Proof without words [asy][asy] real a=sqrt(2); draw(shift(-1,-1)*scale(2)*unitsquare); draw(Circle((0,0),1)); draw(rotate(45,(0,0))*shift(-a/2,-a/2)*scale(a)*unitsquare); draw((1,0)--(-1,0)^^(0,1)--(0,-1),dashed); draw((1,1)--(-1,-1)^^(1,-1)--(-1,1),dashed); [/asy][/asy]
19.09.2020 04:21
Well clearly, the answer is $\text{reciprocal}(\frac{1}{2})$. [asy][asy] draw((0, 0) -- (0, 2) -- (2, 2) -- (2, 0) -- (0, 0)); draw((0, 1) -- (2, 1)); draw((1, 0) -- (1, 2)); draw((0, 1) -- (1, 2) -- (2, 1) -- (1, 0) -- (0, 1) ); [/asy][/asy]
19.09.2020 04:24
19.09.2020 04:26
I'm incredibly surprised that all of you agreed on an incorrect answer. Read the problem again.
19.09.2020 04:30
asbodke wrote: I'm incredibly surprised that all of you agreed on an incorrect answer. Read the problem again. Big oops lol
19.09.2020 04:32
Yes, it is $\frac{2}{1}$, remember the order of the fraction right?
19.09.2020 04:34
lol it makes me look so dumb bc I said "obviously its 1/2"
19.09.2020 04:36
Imagine on the test the answer choices are $A=1/4,B=1/2,C=1,D=2,E=4$ and more that 50% circles $B$ lol
26.11.2020 22:49
Legit this was one my biggest AMC 8 test fears this year. When I saw that percent question with the 15, 20, $x$, and $y$, my heart was beating so fast when I sat that 75 and $133\frac{1}{3}$ were options.