In the figure above, FGHI is inscribed in the circle with ce

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In the figure above, FGHI is inscribed in the circle with center J. What is the ratio of the area of FGHI to the area of the circle?

(1) FGHI is a square.
(2) The area of the circle is 8Ï€.

OA A

Source: Princeton Review

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by fskilnik@GMATH » Thu Jan 17, 2019 4:43 am

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BTGmoderatorDC wrote:Image

In the figure above, FGHI is inscribed in the circle with center J. What is the ratio of the area of FGHI to the area of the circle?

(1) FGHI is a square.
(2) The area of the circle is 8Ï€.
Source: Princeton Review
$$? = {{{S_{{\rm{FGHI}}}}} \over {{S_{{\rm{circle}}}}}}$$
$$\left( 1 \right)\,\,{\rm{FGHI}}\,\,{\rm{square}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,2r = {\rm{FH}} = {\rm{FI}} \cdot \sqrt 2 \hfill \cr
\,\, \Rightarrow \,\,4{r^2} = 2 \cdot {\rm{F}}{{\rm{I}}^{\,2}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = {{{\rm{F}}{{\rm{I}}^{\,2}}} \over {\pi {r^2}}} = {{2{r^2}} \over {\pi {r^2}}}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\pi {r^2} = 8\pi \,\,\,\, \Rightarrow \,\,r > 0\,\,{\rm{unique}}\,\,{\rm{but}}\,\,{\rm{trivial}}\,\,{\rm{geometric}}\,\,{\rm{bifurcation}}\,\,\left( {{\rm{FGHI}}\,\,{\rm{free}}} \right)$$


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