Question Statement

Hira is designing a triangular section of a roof with vertices at points $P(4,1)$, $Q(9,5)$ and $R(7,10)$. She needs to calculate the area of the section to determine how much roofing material is required. Find the area calculated by her.

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Background and Explanation

To find the area of a triangle when the coordinates of all three vertices are known, we can use the determinant method (also called the shoelace formula). This method arranges the coordinates in a $3\times 3$ matrix with an additional column of 1s, allowing us to compute the area directly from the determinant value.

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Solution

First, let's visualize the triangular section of the roof:

To calculate the area of triangle $PQR$ with vertices $P(4,1)$, $Q(9,5)$, and $R(7,10)$, we use the determinant formula for the area of a triangle:

$\text{Area}=\frac{1}{2}\left|\begin{array}{ccc}4& 1& 1\\ 9& 5& 1\\ 7& 10& 1\end{array}\right|$

We expand this determinant along the first row. Recall that the cofactor expansion alternates signs starting with positive for the first element:

$=\frac{1}{2}\left[4\left|\begin{array}{cc}5& 1\\ 10& 1\end{array}\right|-1\left|\begin{array}{cc}9& 1\\ 7& 1\end{array}\right|+1\left|\begin{array}{cc}9& 5\\ 7& 10\end{array}\right|\right]$

Now we calculate each $2\times 2$ determinant:

• First minor: $(5\times 1-10\times 1)=(5-10)=-5$
• Second minor: $(9\times 1-7\times 1)=(9-7)=2$
• Third minor: $(9\times 10-7\times 5)=(90-35)=55$

Substituting these values back:

$=\frac{1}{2}[4(-5)-1(2)+1(55)]$

Simplify the expression inside the brackets:

$=\frac{1}{2}[-20-2+55]$

$=\frac{1}{2}(33)$

$=16.5\text{ sq. units}$

Therefore, the area of the triangular roof section is 16.5 square units.

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Key Formulas or Methods Used

• Determinant method for area of triangle: $\text{Area}=\frac{1}{2}\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|$
• Cofactor expansion: Expanding a $3\times 3$ determinant along the first row using alternating signs $(+-+)$
• Determinant of $2\times 2$ matrix: $\left|\begin{array}{cc}a& b\\ c& d\end{array}\right|=ad-bc$

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Summary of Steps

1. Identify coordinates: Note the vertices $P(4,1)$, $Q(9,5)$, and $R(7,10)$
2. Set up the matrix: Create a $3\times 3$ determinant with $x$-coordinates in column 1, $y$-coordinates in column 2, and 1s in column 3
3. Expand the determinant: Apply cofactor expansion along the first row, calculating the three $2\times 2$ minors with appropriate signs $(+4,-1,+1)$
4. Calculate minors: Compute $(5-10)=-5$, $(9-7)=2$, and $(90-35)=55$
5. Evaluate: Compute $\frac{1}{2}[4(-5)-2+55]=\frac{1}{2}(33)=16.5$ square units