Question

If the area of the triangle with vertices $(-2,0),(0,4)$ and $(0,k)$ is $4$ square units, find the values of $k$ using determinants.

Answer 1

Area of triangle if vertices $A(x_1,y_1),B(x_2,y_2),C(x_3,y_3)$ are given

$△=\frac{1}{2}\left|\begin{array}{ccc}1& x_1& y_1\\ 1& x_2& y_2\\ 1& x_3& y_3\end{array}\right|$

$△=\frac{1}{2}\left|\begin{array}{ccc}1& -2& 0\\ 1& 0& 4\\ 1& 0& k\end{array}\right|$

$4=\frac{1}{2}|2k-8|$

$\Rightarrow 8=|2k-8|$

$\Rightarrow 2k-8=\pm 8$

$\Rightarrow k=8,0$