Answer the following questions with reference to a triangle whose vertices are

MATHEMATICS · Multi-part · CBSE · Mathematics · Coordinate Geometry · Coordinate Geometry

Problem

Answer the following questions with reference to a triangle

ABC

whose vertices are

A(x_1,y_1)

B(x_2,y_2)

and

C(x_3,y_3)

, and an arbitrary point

P(x_p,y_p)

in the plane. (i) Write the algebraic expression for the area of

\triangle ABC

in terms of the coordinates of its vertices. (ii) The point

P

satisfies

\text{Area}(PAB)+\text{Area}(PBC)+\text{Area}(PCA)=\text{Area}(ABC).

Explain, using area considerations only, what this equality tells about the position of

P

with respect to

\triangle ABC

. (iii) State, with justification, what conclusion can be drawn about the position of

P

if instead

\text{Area}(PAB)+\text{Area}(PBC)+\text{Area}(PCA) > \text{Area}(ABC).