91Ó°ÊÓ

The area of triangle having vertices at $\left(a,b\right)$, $\left(c,d\right)$and $\left(e,f\right)$is $\left|A\right|$, where $A=\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ c& d& 1\\ e& f& 1\end{array}\right|$.

Here vertices of the triangle are $\left(a,b\right)=\left(0,0\right)$, $\left(c,d\right)=\left(7,5\right)$and $\left(e,f\right)=\left(2.5,10\right)$. So, substitute 0 for a, 0 for b, 7 for c, 5 for d, 2.5 for e and 10 for f into the expression $A=\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ c& d& 1\\ e& f& 1\end{array}\right|$.

$A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 7& 5& 1\\ 2.5& 10& 1\end{array}\right|$

Find the determinant in this case by using expansion by minors.

$\begin{array}{c}A=\frac{1}{2}\left|\begin{array}{ccc}0& 0& 1\\ 7& 5& 1\\ 2.5& 10& 1\end{array}\right|\\ =\frac{1}{2}\left(\left(0\right)\left|\begin{array}{cc}5& 1\\ 10& 1\end{array}\right|−\left(0\right)\left|\begin{array}{cc}7& 1\\ 2.5& 1\end{array}\right|+\left|\begin{array}{cc}7& 5\\ 2.5& 10\end{array}\right|\right)\end{array}$

The determinant of second order matrix is found by calculating the difference of the product of the two diagonals, that is., $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad−bc$.

Apply this definition to find the $2×2$determinant.

$\begin{array}{c}A=\frac{1}{2}\left(0−0+\left[7\left(10\right)−5\left(2.5\right)\right]\right)\\ =\frac{1}{2}\left(70−\text{12.5}\right)\\ =\frac{1}{2}\left(\text{57.5}\right)\\ =\text{28.75}\end{array}$

Since the area of the triangle is an absolute value, therefore, are of the triangle is 28.75 square units.

Since the area of the triangle is formed by three vertices is 28.75 square units and it is given that 1 unit = 10 miles, therefore, the area of sales territory will be

$⇒28.75×10=287.5$square miles.

Therefore, area is 287.5 square miles.