Question Statement

A landscaper is designing a triangular garden bed with vertices $A(1,4)$, $B(5,1)$ and $C(8,6)$. Calculate the cost of planting mango trees in the garden @Rs. 70 per square unit.

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Background and Explanation

This problem requires finding the area of a triangle given its vertices in the coordinate plane, then applying a unit rate to determine total cost. The determinant method (also known as the shoelace formula) provides a straightforward way to calculate the area using the coordinates directly.

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Solution

First, we find the area of the triangular garden using the determinant method. For a triangle with vertices $(x_1,y_1)$, $(x_2,y_2)$, and $(x_3,y_3)$, the area formula is:

$\text{Area}=\frac{1}{2}\left|\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|\right|$

Substituting the given coordinates $A(1,4)$, $B(5,1)$, and $C(8,6)$:

$\begin{array}{rl}\text{Area of }△ABC& =\frac{1}{2}\left|\begin{array}{ccc}1& 4& 1\\ 5& 1& 1\\ 8& 6& 1\end{array}\right|\\ & =\frac{1}{2}\left[1\left|\begin{array}{cc}1& 1\\ 6& 1\end{array}\right|-4\left|\begin{array}{cc}5& 1\\ 8& 1\end{array}\right|+1\left|\begin{array}{cc}5& 1\\ 8& 6\end{array}\right|\right]\\ & =\frac{1}{2}[1(1-6)-4(5-8)+1(30-8)]\\ & =\frac{1}{2}[1(-5)-4(-3)+1(22)]\\ & =\frac{1}{2}[-5+12+22]\\ & =\frac{1}{2}(29)\\ & =\frac{29}{2}\text{ sq. units}\end{array}$

Now we calculate the cost of plantation at Rs. 70 per square unit:

$\text{Cost}=70\times \frac{29}{2}=35\times 29=1015\text{ rupees}$

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Key Formulas or Methods Used

• Determinant method for area of a triangle: $\text{Area}=\frac{1}{2}\left|\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|\right|$
• Cofactor expansion: Expanding the $3\times 3$ determinant along the first row using $2\times 2$ minors
• Unitary method for cost: $\text{Total Cost}=\text{Area}\times \text{Rate per square unit}$

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Summary of Steps

1. Identify the coordinates of the three vertices: $A(1,4)$, $B(5,1)$, $C(8,6)$
2. Set up the $3\times 3$ determinant matrix with $x$-coordinates, $y$-coordinates, and a column of 1s
3. Expand the determinant along the first row, calculating each $2\times 2$ minor:
   • First minor: $(1\times 1-6\times 1)=-5$
   • Second minor: $(5\times 1-8\times 1)=-3$
   • Third minor: $(5\times 6-8\times 1)=22$
4. Evaluate: $\frac{1}{2}[1(-5)-4(-3)+1(22)]=\frac{1}{2}(29)=\frac{29}{2}$ sq. units
5. Calculate total cost: $70\times \frac{29}{2}=\text{Rs. }1015$