Question

For what value of $x$ are the points $(2,5),(x,2)$ and $(4,3)$ collinear?

Answer 1

Area of the triangle formed by given points

$=\frac{1}{2}|2(3-2)+x(3-5)+4(5-2)|$

$=\frac{1}{2}|2-2x+12|$

$=\frac{1}{2}|14-2x|$

The given points are collinear if and only if area of triangle$=0$

$\Rightarrow \frac{1}{2}|2x-14|=0$

$\Rightarrow (x-7)=0$

$\Rightarrow x=7$

$\therefore$ the required value of $x=7$