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)
\[\text{ Area of the triangle }= ∆ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 3 & - 6 & 1 \\ 7 & 2 & 1\end{vmatrix}\] 
\[ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 6 & - 11 & 0 \\ 7 & 2 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\] 
\[ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 6 & - 11 & 0 \\ 10 & - 3 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\] 
\[ = \frac{1}{2}\begin{vmatrix}6 & - 11 \\ 10 & - 3\end{vmatrix}\] 
\[ = \frac{1}{2}\left( - 18 + 110 \right)\] 
\[ = 46 \text{ square units }\]