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Question
Using determinants, find the area of the triangle with vertices (−3, 5), (3, −6), (7, 2).
Solution
Given:
Vertices of triangle: (− 3, 5), (3, − 6) and (7, 2)
\[\text{ Area of the triangle }= ∆ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 3 & - 6 & 1 \\ 7 & 2 & 1\end{vmatrix}\]
\[ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 6 & - 11 & 0 \\ 7 & 2 & 1\end{vmatrix} \left[\text{ Applying }R_2 \to R_2 - R_1 \right]\]
\[ = \frac{1}{2}\begin{vmatrix}- 3 & 5 & 1 \\ 6 & - 11 & 0 \\ 10 & - 3 & 0\end{vmatrix} \left[\text{ Applying }R_3 \to R_3 - R_1 \right]\]
\[ = \frac{1}{2}\begin{vmatrix}6 & - 11 \\ 10 & - 3\end{vmatrix}\]
\[ = \frac{1}{2}\left( - 18 + 110 \right)\]
\[ = 46 \text{ square units }\]
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