Question

Show that the following sets of points are collinear.

(a) (2, 5), (4, 6) and (8, 8)

(b) (1, −1), (2, 1) and (4, 5)

Answer 1

The formula for the area ‘A’ encompassed by three points, and is given by the formula,

We know area of triangle formed by three points is given by

If three points are collinear the area encompassed by them is equal to 0.

The three given points are A(2, 5), B(4, 6) and C(8, 8). Substituting these values in the earlier mentioned formula we have,
$$\begin{gathered} A=\frac{1}{2}\left|2\left(6-8\right)+4\left(8-5\right)+8\left(5-6\right)\right| \\ =\frac{1}{2}\left|2\left(-2\right)+4\left(3\right)+8\left(-1\right)\right| \\ =\frac{1}{2}\left|-4+12-8\right| \\ =\frac{1}{2}\left|-12+12\right| \\ =0 \end{gathered}$$

Since the area enclosed by the three points is equal to 0, the three points need to be.

The three given points are A(1, −1), B(2, 1) and C(4, 5). Substituting these values in the earlier mentioned formula we have,
$$\begin{gathered} A=\frac{1}{2}\left|1\left(1-5\right)+2\left(5+1\right)+4\left(-1-1\right)\right| \\ =\frac{1}{2}\left|1\left(-4\right)+2\left(6\right)+4\left(-2\right)\right| \\ =\frac{1}{2}\left|-4+12-8\right| \\ =\frac{1}{2}\left|-12+12\right| \\ =0 \end{gathered}$$

Since the area enclosed by the three points is equal to 0, the three points need to be.