Question

Find values of $k$ if area of triangle is $4$ square units and vertices are
$\left(i\right)$ $(k,0),(4,0)$ and $(0,2)$
$\left(ii\right)$ $(-2,0),(0,4)$ and $(0,k)$

Answer 1

$\left(i\right)$ Given vertices: $(k,0),(4,0)$ and $(0,2)$
Area of triangle is $4$ square units.
The area is always positive but, $\Delta$ can have both positive and negative signs.
$\therefore \Delta =\pm 4$
The area of trinage is given by
$\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
$\Rightarrow \pm 4=\frac{1}{2}\left|\begin{array}{ccc}k& 0& 1\\ 4& 0& 1\\ 0& 2& 1\end{array}\right|$
$\Rightarrow \pm 4=\frac{1}{2}(k\left|\begin{array}{cc}0& 1\\ 2& 1\end{array}\right|-0\left|\begin{array}{cc}4& 1\\ 0& 1\end{array}\right|+1\left|\begin{array}{cc}4& 0\\ 0& 2\end{array}\right|)$
$\Rightarrow \pm 4\times 2=\left(k\right(-2)-0+1(8\left)\right)$
$\Rightarrow \pm 8=-2k+8$
So, $8=-2k+8$ or $-8=-2k+8$
$\Rightarrow 8=-2k+8$
$\Rightarrow 8-8=-2k$
$\Rightarrow 0=-2k$
$\Rightarrow k=0$
$\Rightarrow -8=-2k+8$
$\Rightarrow -8-8=-2k$
$\Rightarrow -16=-2k$
$\Rightarrow k=\frac{-16}{-2}=8$
So, the required value of $k$ is $8$ or $0$.

$\left(ii\right)$ Given vertices: $(-2,0),(0,4)$ and $(0,k)$
Area of triangle is $4$ square units.
The area is always positive but, $\Delta$ can have both positive and negative signs.
$\therefore \Delta =\pm 4$
The area of triangle is given by
$\Delta =\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
$\Rightarrow \pm 4=\frac{1}{2}\left|\begin{array}{ccc}-2& 0& 1\\ 0& 4& 1\\ 0& k& 1\end{array}\right|$
$\Rightarrow \pm 4=\frac{1}{2}(-2\left|\begin{array}{cc}4& 1\\ k& 1\end{array}\right|-0\left|\begin{array}{cc}0& 1\\ 0& 1\end{array}\right|+1\left|\begin{array}{cc}0& 4\\ 0& k\end{array}\right|)$
$\Rightarrow \pm 4=\frac{1}{2}(-2(4-k)-0(0-0)+1(0-0\left)\right)$
$\Rightarrow \pm 4=\frac{1}{2}(-2(4-k\left)\right)$
$\Rightarrow \pm 4=-1(4-k)$
$\Rightarrow \pm 4=-4+k$
So, $4=-4+k$ or $-4=-4+k$
$\Rightarrow 4=-4+k$
$\Rightarrow 4+4=k$
$\Rightarrow k=8$
$\Rightarrow -4=-4+k$
$\Rightarrow k=-4+4$
$\Rightarrow k=0$
So, the required value of $k$ is $k=8$ or $k=0$.