Question

Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).

Answer 1

Given:
Vertices of triangle: (− 3, 5), (3, − 6) and (7, 2)

$$\begin{gathered} \mathrm{Area}\mathrm{of}\mathrm{the}\mathrm{triangle}=∆=\frac{1}{2}\left|\begin{array}{ccc}-3& 5& 1\\ 3& -6& 1\\ 7& 2& 1\end{array}\right| \\ =\frac{1}{2}\left|\begin{array}{ccc}-3& 5& 1\\ 6& -11& 0\\ 7& 2& 1\end{array}\right|\left[\mathrm{Applying}{R}_2\to R_2-R_1\right] \\ =\frac{1}{2}\left|\begin{array}{ccc}-3& 5& 1\\ 6& -11& 0\\ 10& -3& 0\end{array}\right|\left[\mathrm{Applying}{R}_3\to R_3-R_1\right] \\ =\frac{1}{2}\left|\begin{array}{cc}6& -11\\ 10& -3\end{array}\right| \\ =\frac{1}{2}\left(-18+110\right) \\ =46\mathrm{square}\mathrm{units} \end{gathered}$$