Question

Prove (by showing that the area of the triangle formed by them is zero) that the following sets of three points are in a straight line:

$(1,4),(3,-2),$ and $(-3,16)$

Answer 1

Area of triangle having angular points
$(x_1,y_1)$ $(x_2,y_2)$ $(x_3,y_3)$ is given by the formula

$△=\frac{1}{2}|x_1{y}_2+x_2{y}_3+x_3{y}_1-x_2{y}_1-x_3{y}_2-x_1{y}_3|$

$△=\frac{1}{2}|1\times (-2)+3\times 16+(-3)\times 4-3\times 4-(-3\left)\right(-2)-1\times 16|$

$\Rightarrow △=0$

$\Rightarrow$ All three points are in a straight line.