fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 15)
If quadrilaterals ABCD and ADEF have the same areas, is AF > 28 ?
(1) EF = 10
(2) CE = 24
$${S_{{\mathop{\rm ABCD}\nolimits} }} = {S_{{\rm{ADEF}}}} = {S_{{\rm{ADEG}}}} + {S_{{\rm{AGF}}}}\,\,\,\,\,\left( * \right)$$
$$AF\,\,\mathop > \limits^? \,\,28$$
$$\left( 1 \right)\,\,\left\{ \matrix{
\,{\rm{image}}\,\,{\rm{left}}:\,\,\left( * \right)\,\,4 \cdot 6 = 3 \cdot 6 + {{3 \cdot 4} \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{viable}}\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{image}}\,\,{\rm{right}}:\,\,\left( * \right)\,\,x \cdot 6 = 28 \cdot 6 + {{28 \cdot 4} \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{viable}}\,\,\,\left( {x = {{28 \cdot 8} \over 6} > 0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
$$\left( 2 \right)\,\,\left\{ \matrix{
\,{\rm{image}}\,\,{\rm{left}}:\,\,\left( * \right)\,\,18 \cdot y = 6 \cdot y + {{6 \cdot 4} \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{viable}}\,\,\,\,\left( {y = 1 > 0} \right) \Rightarrow \,\,\,\,\left\langle {{\rm{NO}}} \right\rangle \hfill \cr
\,{\rm{image}}\,\,{\rm{right}}:\,\,\left( * \right)\,\,18 \cdot y = 6 \cdot y + {{6 \cdot 28} \over 2}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{viable}}\,\,\,\left( {y = 7 > 0} \right)\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\rm{YES}}} \right\rangle \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,{\rm{Extremal}}\,\,{\rm{Scenario}}\,\,\,\left( {\left( * \right)\,\,{\rm{ignored}}} \right)\,\,:\,\,A{F_{\,{\rm{ExtScen}}}}\,\,\mathop < \limits^{{\rm{near}}} \,\,13 \cdot 2 = 26$$
$$ \Rightarrow \,\,AF < \,\,A{F_{\,{\rm{ExtScen}}}} < 28\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.$$
The correct answer is (C).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
POST-MORTEM: (1+2) It is possible to prove that AF < 16. See:  <span text-bold"><a href="https://gmatclub.com/forum/if-quadrilat ... l#p2243965" target="_blank" class="text-info">Shobhit7´s argument</a></span>
