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

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BTGmoderatorDC: Moderator; Posts: 7187; Joined: Thu Sep 07, 2017 4:43 pmFollowed by: 23 members

by BTGmoderatorDC » Thu Jan 17, 2019 4:15 am

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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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Source: Beat The GMAT — Data Sufficiency | Thu Jan 17, 2019 4:15 am

fskilnik@GMATH: GMAT Instructor; Posts: 1449; Joined: Sat Oct 09, 2010 2:16 pmFollowed by: 33 members; Thanked: 59 times

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

BTGmoderatorDC wrote:

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

$$? = {{{S_{{\rm{FGHI}}}}} \over {{S_{{\rm{circle}}}}}}$$
$$\left( 1 \right)\,\,{\rm{FGHI}}\,\,{\rm{square}}\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,2r = {\rm{FH}} = {\rm{FI}} \cdot \sqrt 2 \hfill \cr
\,\, \Rightarrow \,\,4{r^2} = 2 \cdot {\rm{F}}{{\rm{I}}^{\,2}} \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,? = {{{\rm{F}}{{\rm{I}}^{\,2}}} \over {\pi {r^2}}} = {{2{r^2}} \over {\pi {r^2}}}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,\pi {r^2} = 8\pi \,\,\,\, \Rightarrow \,\,r > 0\,\,{\rm{unique}}\,\,{\rm{but}}\,\,{\rm{trivial}}\,\,{\rm{geometric}}\,\,{\rm{bifurcation}}\,\,\left( {{\rm{FGHI}}\,\,{\rm{free}}} \right)$$

We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.

Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br

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