Answer:
k = 5 or k = -19
Step-by-step explanation:
To find the area of a triangle formed by three points A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) using determinants, we can use the formula:
[tex]\textsf{Area of $\triangle ABC$}=\dfrac{1}{2}\left|\begin{array}{ccc}x_1&y_1&1\\x_2&y_2&1\\x_3&y_3&1\end{array}\right|[/tex]
This expands to:
[tex]\textsf{Area of $\triangle ABC$}=\dfrac{1}{2}\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right|[/tex]
In this case, the area of the triangle is 12 square units, and the points of the triangle are A(-3, 6), B(-4, 4) and C(k, -2).
Substitute these values into the formula:
[tex]12=\dfrac{1}{2}\left|(-3)(4-(-2))+(-4)(-2-6)+k(6-4)\right|[/tex]
Simplify:
[tex]12=\dfrac{1}{2}\left|(-3)(6)+(-4)(-8)+k(2)\right|[/tex]
[tex]12=\dfrac{1}{2}\left|-18+32+2k\right|[/tex]
[tex]12=\dfrac{1}{2}\left|14+2k\right|[/tex]
Multiply both sides by 2 to isolate the absolute value expression:
[tex]\left|14+2k\right|=24[/tex]
Set the expression within the absolute value equal to both the positive and negative values of the right side of the equation, and then solve both for k:
[tex]\begin{aligned}14+2k&=24\\2k&=10\\k&=5\end{aligned}[/tex]
[tex]\begin{aligned}14+2k&=-24\\2k&=-38\\k&=-19\end{aligned}[/tex]
Therefore, the value of k is:
[tex]\Large\boxed{\boxed{k = 5\;\;\textsf{or}\;\;k = -19}}[/tex]
