Data Sufficiency

Source: Princeton Review

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

The OA is B.

If p, q and r are all positive values and AO and AB is the area of triangular region ABO less than 12

$$p,q,r\ \ >0\ \ and\ AO\ >AB>$$
$$p^2+q^2>>\left(p-r\right)^2+q^2=2pr>r^2$$
$$Thus\ r>0\ and\ 2p>r$$
$$Area\ of\ triangle\ ABO=\frac{1}{2}\cdot base\cdot height$$
where base = r and height = q
$$Area=0.5\cdot r\cdot q=0.5rq$$
from question, we need to verify if
$$\frac{0.5rq}{0.5}=\frac{12}{0.5}$$
$$rq<24$$

Statement 1
r=7 $$7q<24,\ no\ \inf ormation\ about\ q,$$
$$hence\ statement\ 1\ is\ INSUFFICIENT$$

Statement 2
$$p=4\ and\ q\ =3$$
$$2p>r$$
$$if\ we\ will\ have\ 2\cdot4=r$$
$$8>r\ hence\ r<8$$
Thus the value of rq must be less than 24 as q=3 and r=8
from rq<24 whatever combination of numbers r and q takes it must be always be less than 24 which in turn means that area of triangular region ABO is less than 12 $$statement\ 2\ above\ is\ INSUFFICIENT\$$

$$answer\ is\ option\ B$$

BTGmoderatorLU wrote:Source: Princeton Review

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

$${S_{\Delta ABO}} = \frac{{r \cdot q}}{2}\,\,\mathop < \limits^? \,\,12\,\,\,\,\,\, \Leftrightarrow \,\,\,\boxed{\,\,r \cdot q\,\,\mathop < \limits^? \,\,24\,\,}$$



$$\left( 1 \right)\,\,r = 7\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,q = 1\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{YES}}} \right\rangle \,\, \hfill \cr
\,{\rm{Take}}\,\,q = 4\,\,\,\, \Rightarrow \,\,\,\left\langle {{\rm{NO}}} \right\rangle \,\, \hfill \cr} \right.$$

$$\left( 2 \right)\,\,\left( {p,q} \right) = \left( {4,3} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,r \cdot q < 8 \cdot 3\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle $$
$$\left( * \right)\,\,4 = p > \frac{r}{2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,r < 8$$

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.