Answer
$\sqrt{11}$
Work Step by Step
$u=(0,1,2)-(2,-3,4)=(-2,4,-2)$
$v=(-1,2,0)-(2,-3,4)=(-3,5,-4)$
are two sides of the triangle.
We know that for a matrix
$
\left[\begin{array}{rrr}
a & b & c \\
d &e & f \\
g &h & i \\
\end{array} \right]
$
the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$
$u×v$ is the determinant of the matrix $\begin{bmatrix}
i& j & k \\
-2& 4&-2\\
-3&5 &-4 \\
\end{bmatrix}
$
Thus $u×v=(-6,2,2).$
Thus $A=0.5|u\times v|=0.5\sqrt{(-6)^2+2^2+2^2}=0.5\sqrt{44}=\sqrt{11}$
