Pre calculus question !!!

Question

The vertices of a triangle ΔABC are A( p,q) , B(r, s) ,C(u, v) . Assume that these coordinates satisfy 2p+3r+4u=0   ,  2q + 3s + 4v=0     , and the origin O(0,0) lies in the interior of ΔABC . If (area of ΔOBC )=r (area of ΔABC ), then r = 

 

(a) 2\9    (b) 4\9    (c) 2\7    (d) 3\7    (e) 4\7

Answer 1

Hi again, Bader.

I do not know if there is an easier way to do this, but....

 

I set up a triangle whose vertices satisfied the above formulas (2p+3r+4u=0 and 2q+3s+4v=0) and had the origin in the interior.  Then I found the area of that triangle.  I made the triangle with the origin and found its area.  Then I took the 

 

Area of big triangle (ABC)  (I got 135 with my choice of A, B and C)

---------------------------------------------------------------------------  = r = 2/9

Area of small triangle (OBC) (I got 30 with my choice of A, B and C)

 

Your teacher can probably show you a better way.  My way took a long time and a lot of calculations.