Answer
$\frac{51}{2}$
Work Step by Step
Step 1. Identify the coordinates of the vertices of the triangle:
$(-4,1), (1,5),(5,-2)$
Step 2. Use the formula (in section 10.6) to set up the determinant:
$\begin{array} \\A=\pm\frac{1}{2}\\ \end{array}
\begin{vmatrix} -4&1&1\\ 1&5&1\\5&-2&1\end{vmatrix}$
Step 3. Perform the row operations $R_2-R_1\to R_2$ and $R_3-R_1\to R_3$
$\begin{array} \\A=\pm\frac{1}{2}\\ \end{array}
\begin{vmatrix} -4&1&1\\ 5&4&0\\9&-3&0\end{vmatrix}$
Step 4. Evaluate the determinate using column-3 expansion:
$A=\pm\frac{1}{2}(-5\times3-4\times9)=\pm\frac{51}{2}$
Step 5. Conclusion: the area of the triangle is $\frac{51}{2}$
