Question

if the vertices of triangle ABC are A(0,-6) B(2,5) C(-1,3) and P(x,y) is a point in the interior of triangle ABC ,show that the ratio of area of triangle PBC to triangle ABC is (2x-3y+11)/49​

Answer 1

The ratio of area of triangle Δ PBC to triangle Δ ABC is $\frac{2x - 3y + 11}{29}$.

Step-by-step explanation:

The triangle Δ ABC has vertices A(0,-6), B(2,5) and C(-1,3) and P(x,y) is a point in the interior of the triangle Δ ABC.

Now, the area of the Δ ABC = $\frac{1}{2}|0(5 - 3) + 2(3 + 6) - 1(- 6 - 5)| = \frac{29}{2}$ sq. units.

And the area of the Δ PBC = $\frac{1}{2}|x(5 - 3) + 2(3 - y) - 1(y - 5)| = \frac{|2x - 3y + 11|}{2}$ sq. units.

Therefore, the ratio of the area of triangle Δ PBC to triangle Δ ABC is $\frac{2x - 3y + 11}{29}$ (Answer)