Finding the area of a triangle, given three coordinates?

Question

I have been attempting the following question:

I can get the answer to part (a) and part (b) but am struggling to find the area of the triangle from the three coordinates. Here is my working:

I have tried showing he triangle is right-angled, and have used Heron’s formula, but I just can’t seem to get the correct answer. Any help would be greatly appreciated!

Edit: I have now applied the shoelace method (pictured below) and have come to the correct answer:

Answer 1

Once you’ve got the vertices of the triangle, its area can be computed via a determinant: $$\frac12 \begin{vmatrix}P-R\\Q-R\end{vmatrix} = \frac12\begin{vmatrix}x_P&y_P & 1\\x_Q&y_Q & 1\\x_R&y_R & 1\end{vmatrix}.$$ You can view the first determinant as $\lvert(P-R)\times(Q-R)\rvert$, although strictly speaking the cross product isn’t defined in $\mathbb R^2$.