Question

Determine whether the point (-3, 2) lies inside or outside the triangle whose sides are given by the equations $x+y-4=0$, $3x-7y+8=0$, $4x-y-31=0$.

Answer 1

Let ABC be the triangle of sides AB, BC and CA, whose equations $x+y-4=0$, $3x-7y+8=0$ and $4x-y-31=0$, respectively.

On solving them, we get A(7, -3), B(2, 2) and C(9, 5) as the coordinates of the vertices.

Let P(-3, 2) be the given point.

The given point P(-3, 2) will lie inside the triangle ABC, if

(i) A and P lies on the same side of BC

(ii) B and P lies on the same side of AC

(iii) C and P lies on the same side of AB

Thus, if A and P lie on the same side of BC, then

(21 + 21 + 8)(-9 -14 + 8)>0

$\Rightarrow 50\times (-15)>0$

$\Rightarrow -750>0$, which is false.

Therefore, the point (-3, 2) lies outside triangle ABC.