Intermediate Algebra, Chapter 3, 3.2, Section 3.2, Problem 100

Three points that lie on the same straight line are said to be collinear. Consider the points $A(3,1), B(6,2)$ and $C(9,3)$. Use the slope formula to determine whether the point $(0,6), (4,-5)$ and $(-2,12)$.

Let $A$ be the point $(0,6)$

$\phantom{Let}$ $B$ be the point $(4,-5)$

$\phantom{Let}$ $C$ be the point $(-2,12)$

Slope of segment $AB$

$\displaystyle m_{AB} = \frac{-5-6}{4-0} = - \frac{11}{4}$

Slope of segment $BC$

$\displaystyle m_{BC} = \frac{12-(-5)}{-2-4} = - \frac{17}{6}$

Slope of segment $AC$

$\displaystyle m_{AC} = \frac{12-6}{-2-0} = - \frac{6}{2} = -3$

The slope of each segment are not the same. Thus, these three points cannot be collinear.