Question

If $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ are the vertices of a triangle, then the equation $\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|+\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_3& y_3& 1\end{array}\right|=0$ represents

• A. the median through A
• B. the altitude through A
• C. the perpendicular bisector of BC
• D. the line joining the centroid with a vertex

Answer 1

Answer: (A), (D)

The correct options are
A the median through A
D the line joining the centroid with a vertex

Equation of the line passing though two given points $(x_1,y_1)$ and $(x_2,y_2)$ is $\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|=0$.
Now, the given expression is
$\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_2& y_2& 1\end{array}\right|+\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_3& y_3& 1\end{array}\right|=0$
$\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ x_2+x_3& y_2+y_3& 2\end{array}\right|=0$ or
$\left|\begin{array}{ccc}x& y& 1\\ x_1& y_1& 1\\ \frac{x_2+x_3}{2}& \frac{y_2+y_3}{2}& 1\end{array}\right|=0$
This is the equation of the line passing through the points $(x_1,y_1)$ and $\left(\frac{x_2+x_3}{2},\frac{y_2+y_3}{2}\right).$ This is an equation of median through vertex $A$.
Hence, options $A$ and $D$ are the correct answers.