Explanation

(d)
(1) We have, $\left| \begin{matrix}
x_{1} & y_{1} & 1 \\
x_{2} & y_{2} & 1 \\
x_{3} & y_{3} & 1
\end{matrix} \right| = \left| \begin{matrix}
a_{1} & b_{1} & 1 \\
a_{2} & b_{2} & 1 \\
a_{3} & b_{3} & 1
\end{matrix} \right|$
This shows that the area of both the triangles are same. But the equality of the areas of the triangles do not ensure the congruence of the triangle
(2) The equation of a line passing through the origin is $y = mx$. If it is equidistant from the points $A(2,\ 2)$ and $B(4,\ 0)$, then
$\left| \frac{2m - 2}{\sqrt{m^{2} + 1}} \right| = \left| \frac{4m - 0}{\sqrt{m^{2} + 1}} \right|$
$\Rightarrow (2m - {2)}^{2} = ({4m)}^{2}$
$\Rightarrow (m - {1)}^{2} = {4m}^{2}$
$\Rightarrow \ {3m}^{2} + 2m - 1 = 0$
$\Rightarrow m = \frac{1}{3},\ - 1$
Hence, there are two lines $y = \frac{x}{3}$ and $y = - x$ passing through the origin and equidistance from $A(2,\ 2)$ and $B(4,\ 0)$
Hence, both of these two statements are not correct