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.
