Q43. - Algebra 2

Question

During an archaeological dig, a coordinate grid is laid over the site to identify the location of artifacts as they are excavated. During a dig, three corners of a rectangular building have been partially unearthed at (-1,6), (4,5), and (-3, -4). If each square on the grid measures one square foot, estimate the area of the floor of the building.

Step-by-Step Solution

Answer

Area of the rectangle is 52 square units.

Step 1 - Define area of triangle

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 $\left(a,b\right)=\left(-1,6\right)$, $\left(c,d\right)=\left(4,5\right)$ and $\left(e,f\right)=\left(-3,-4\right)$. So, substitute -1 for a, 6 for b, 4 for c, 5 for d, -3 for e and -4 for f into the expression $A=\frac{1}{2}\left|\begin{array}{ccc}a& b& 1\\ c& d& 1\\ e& f& 1\end{array}\right|$.

$A=\frac{1}{2}\left|\begin{array}{ccc}-1& 6& 1\\ 4& 5& 1\\ -3& -4& 1\end{array}\right|$

Step 2 - Find the determinant

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

 

$\begin{array}{c}A=\frac{1}{2}\left|\begin{array}{ccc}-1& 6& 1\\ 4& 5& 1\\ -3& -4& 1\end{array}\right|\\ =\frac{1}{2}\left(-1\left|\begin{array}{cc}5& 1\\ -4& 1\end{array}\right|-6\left|\begin{array}{cc}4& 1\\ -3& 1\end{array}\right|+\left|\begin{array}{cc}4& 5\\ -3& -4\end{array}\right|\right)\end{array}$

Step 3 - Evaluate 2 × 2 determinants

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 $2\times 2$ determinant.

 

$\begin{array}{c}A=\frac{1}{2}\left(-\left[5-\left(-4\right)\right]-6\left[4-\left(-3\right)\right]+\left[4\left(-4\right)-5\left(-3\right)\right]\right)\\ =\frac{1}{2}\left(-\left(9\right)-6\left[7\right]+\left[-16+15\right]\right)\\ =\frac{1}{2}\left(-9-42-1\right)\\ =\frac{1}{2}\times \left(-52\right)\\ =-26\end{array}$

 

Since the area of the triangle is an absolute value, therefore, are of the triangle is 26 square units.

Step 4 - Estimate area of rectangle

Since the area of the triangle is formed by three vertices of the rectangle, therefore the triangle must divide the rectangle into half. 

 

Therefore, area of rectangle = $2\times$area of triangle.

 

$\begin{array}{c}\text{area of rectangle}=2\times 26\\ =52\end{array}$

 

Hence, area of the rectangle is 52 square units.