Answer
3 square units.
Work Step by Step
Let the vector connecting the points (1,2) and (3,4) be $\textbf{v}$ and the vector connecting points (3,4) and (-2,2) be $\textbf{w}$. Then,
$\textbf{v}=(3-1)\textbf{i}+(4-2)\textbf{j}=2\textbf{i}+2\textbf{j}$
$\textbf{w}=(-2-3)\textbf{i}+(2-4)\textbf{j}=-5\textbf{i}-2\textbf{j}$
The area of the triangle is given by:
$\frac{||\textbf{v}\times\textbf{w}||}{2}$
$\textbf{v}\times\textbf{w}=\begin{vmatrix}\textbf{i}&\textbf{j}&\textbf{k}\\2&2&0\\-5&-2&0\end{vmatrix}$
$=\textbf{k}(-4+10)=6\textbf{k}$
$||\textbf{v}\times\textbf{w}||=\sqrt {6^{2}}=6$
Area= $\frac{||\textbf{v}\times\textbf{w}||}{2}=\frac{6}{2}=3$
