College Algebra, Chapter 7, 7.4, Section 7.4, Problem 62

An outdoors club is purchasing land to set up a conservation area. The last remaining piece they need to buy is the triangular plot shown in the figure. Use the determinant formula for the area of a triangle to find the area of the plot.







Based from the graph, the triangular plot has a vertices $(1000,2000), (2000,6000)$ and $(5000,4000)$. Using the formula for areas of triangles using determinants, we get


$
\begin{equation}
\begin{aligned}

\text{area } =& \pm \frac{1}{2} \left| \begin{array}{ccc}
1000 & 2000 & 1 \\
2000 & 6000 & 1 \\
5000 & 4000 & 1
\end{array} \right|
\\
\\
=& \pm \frac{1}{2} \left[ 1000 \left| \begin{array}{cc}
6000 & 1 \\
4000 & 1
\end{array} \right| - 2000 \left| \begin{array}{cc}
2000 & 1 \\
5000 & 1
\end{array} \right| + 1 \left| \begin{array}{cc}
2000 & 6000 \\
5000 & 4000
\end{array} \right| \right]
\\
\\
=& \pm \frac{1}{2} \left[ 1000 (6000 \cdot 1 - 1 \cdot 4000) - 2000 (2000 \cdot 1 - 1 \cdot 5000) + (2000 \cdot 4000 - 6000 \cdot 5000) \right]
\\
\\
=& \pm \frac{1}{2} (-14,000,000)

\end{aligned}
\end{equation}
$


To make the area positive, we choose the negative sign in the formula. Thus, the area of the triangular plot is

area = $7,000,000 ft^2$