saurabh87 wrote:
In the xy-plane, point O is located at the origin, point A has coordinates (p,q), and point B has coordinates (r,0). If p, q, and r are all positive values and AO > AB, is the area of triangular region ABO less than 12 ?
(1) r = 7
(2) p = 4 and q = 3
\({S_{\Delta ABO}} = \frac{{r \cdot q}}{2}\,\,\mathop < \limits^? \,\,12\,\,\,\,\,\, \Leftrightarrow \,\,\,\boxed{\,\,r \cdot q\,\,\mathop < \limits^? \,\,24\,\,}\)
\(\left( 1 \right)\,\,r = 7\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,q = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr \\
\,{\rm{Take}}\,\,q = 4\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.\)
\(\left( 2 \right)\,\,\left( {p,q} \right) = \left( {4,3} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,r \cdot q < 8 \cdot 3\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\rm{YES}}} \right\rangle\)
\(\left( * \right)\,\,4 = p > {r \over 2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,r < 8\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.