Q. 58 - Precalculus Enhanced with Graphing Utilities

Question

Geometry: Collinear Points Using the result obtained in Problem 57, show that three distinct points $\left(x_1,y_1\right), \left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$ are collinear (lie on the same line) if and only if

$\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Step-by-Step Solution

Answer

It is proved that the given points $\left(x_1,y_1\right), \left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$ are collinear only if,

$\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Step 1. Given Information

We are given three $\left(x_1,y_1\right), \left(x_2,y_2\right),\text{ and }\left(x_3,y_3\right)$  points.

We have to show that the above points are collinear only if,

$\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Step 2. Proving the collinear points

The equation of a line passing through two points $\left(x_1,y_1\right), \left(x_2,y_2\right)$ is given by,

$\left|\begin{array}{lll}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$

Since, all points lie on the same line, so

$\left|\begin{array}{lll}{x}_3& y_3& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$

or

$\left|\begin{array}{lll}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

Hence Proved.