Question

Prove that the points $(0,5),(0,-9)$ and $(3,6)$ are non-collinear.

Answer 1

Given are coordinates of three points.

To prove: The three points are non-collinear.

Let the given points be $A(0,5),B(0,-9)andC(3,6)$.

We know that, three points are collinear if the area of the triangle they form is $0$. Conversely, if the given points make a triangle having a non-zero area, then they are non-collinear.

We also know that, the area of the triangle formed by points $A(x_1,y_1),B(x_2,y_2)andC(x_3,y_3)$ is:

$\frac{1}{2}\left|x_1\right(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2\left)\right|$

$\therefore$ Area of $△ABC=\frac{1}{2}\left|0\right(-9-6)+0(6-5)+3(5+9\left)\right|$

$\Rightarrow \frac{1}{2}|0+0+42|$

$\Rightarrow \frac{42}{2}=21$

Hence, Area of $△ABC=21$

Since, area of $△ABC\ne 0$, points $A,BandC$ are non-collinear. $\left[Henceproved\right]$