Using slopes, find the value of x for which the points A

For three points to be collinear, the slope of all pairs must be equal, that is the slope of AB = slope of BC = slope of CA

Given points are A(5, 1), B(1, -1) and C(x, 4)

slope $=\left(\frac{y_{2}-y_{1}}{x_{2}-x_{1}}\right)$

Slope of $A B=\left(\frac{-1-1}{1-5}\right)=\frac{-2}{-4}=\frac{1}{2}=0.5$

The slope of $B C=\left(\frac{4+1}{x-1}\right)=\left(\frac{5}{x-1}\right)$

Slope of $C A=\left(\frac{4-1}{x-5}\right)=\left(\frac{3}{x-5}\right)$

The slope of all lines must be the same

$\Rightarrow 0.5=\left(\frac{5}{x-1}\right)$

$\Rightarrow 0.5 x-0.5=5$

$\Rightarrow 0.5 x=5.5$

$\Rightarrow \mathrm{x}=11$

Note:- We can use any two points to get the value of “x”.