Using determinants, find the area of the triangle whose vertices … - Vidyadip

Question

Using determinants, find the area of the triangle whose vertices are $(1, 4), (2, 3)$ and $(-5, -3)$. Are the given points collinear?

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Answer

$\triangle-\frac{1}{2}=\begin{vmatrix}1&4&1\\2&3&1\\-5&-3&1\end{vmatrix}$
$=\frac{1}{2}\begin{vmatrix}1&4&1\\1&-1&0\\-5&-3&1\end{vmatrix}$ [Applying $R_2 → R_2 - R_1$]
$=\frac{1}{2}\begin{vmatrix}1&4&1\\1&-1&0\\-6&-7&0\end{vmatrix}$ [Applying $R_3 → R_3 - R_1$]
$=\frac{1}{2}\begin{vmatrix}1&-1\\-6&-7\end{vmatrix}$
$=\frac{1}{2}(-7-6)$
$=\frac{13}{2}\text{square units}$ $[\because$ Area cannot be negative$]$
Thus, (1, 4), (2, 3) and (-5, 3) are not collinear because $\begin{vmatrix}1&4&1\\2&3&1\\-5&-3&1\end{vmatrix}$ is not equal to zero.

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