Question

Using the method of slope, show that the points A(16, -18), B(3, -6) and C(-10, 6) are collinear.

Answer 1

Hint: Any three points are collinear when the slope of the lines formed by any two pairs of points are equal. That is, mAB=mBC=mAC. The slope between the points (x1, y1) and (x2, y2) can be calculated as-
${\text{m}} = \dfrac{{{{\text{y}}_2} - {{\text{y}}_1}}}{{{{\text{x}}_2} - {{\text{x}}_1}}}$

Complete step-by-step answer:

We will first find the slope mAB, this can be calculated as-
$\begin{gathered} {{\text{m}}_{AB}} = \dfrac{{ - 6 - \left( { - 18} \right)}}{{3 - 16}} \\\ {{\text{m}}_{AB}} = \dfrac{{12}}{{ - 13}} = - \dfrac{{12}}{{13}} \\\ \end{gathered}$

The slope mBC can be calculated as-
$\begin{array}{l} {{\rm{m}}_{BC}} = \dfrac{{6 - \left( { - 6} \right)}}{{ - 10 - 3}}\\\ {{\rm{m}}_{BC}} = \dfrac{{12}}{{ - 13}} = - \dfrac{{12}}{{13}} \end{array}$

Now, AB and BC have the same slope and pass through the same point B, hence they coincide. This means that the points A, B and C lie on the same line and are collinear points.

Note: We can also solve this problem by finding the equation of the line through AB and then check if C satisfies it or not.
The equation of line AB is given by-
y - y1 = m(x - x1)
${\text{y}} + 6 = - \dfrac{{12}}{{13}}\left( {{\text{x}} - 3} \right)$
13y + 78 = -12x + 36
12x + 13y + 42 = 0
Substituting C(-10, 6),
-120 + 78 + 42 = 0
0 = 0
Hence, the points are collinear.