Triangle ABC:
The given triangle has vertices A(4,1), B(-3,2), and C(0, k). We need to find the value of k when the area of the triangle is 12 square units.

Formula for Area of a Triangle:
The area of a triangle can be calculated using the formula: Area = 1/2 * base * height.

Calculating the Base:
To calculate the base of the triangle, we need to find the distance between two points. Let's consider points A(4,1) and B(-3,2).

Using the distance formula, we have:
Base = √((x2 - x1)^2 + (y2 - y1)^2)
= √((-3 - 4)^2 + (2 - 1)^2)
= √((-7)^2 + (1)^2)
= √(49 + 1)
= √50

Calculating the Height:
To calculate the height of the triangle, we need to find the perpendicular distance from vertex C(0, k) to the line AB.

Equation of Line AB:
The equation of a line passing through points A(4,1) and B(-3,2) can be found using the slope-intercept form: y = mx + c.

Slope (m) = (y2 - y1) / (x2 - x1)
= (2 - 1) / (-3 - 4)
= 1 / -7
= -1/7

Using the point-slope form: y - y1 = m(x - x1), we can substitute the values of point A(4,1) to find the equation of line AB:
y - 1 = (-1/7)(x - 4)
7y - 7 = -x + 4
x + 7y = 11

Perpendicular Distance:
The perpendicular distance from a point (x1, y1) to a line Ax + By + C = 0 is given by:
Distance = |Ax1 + By1 + C| / √(A^2 + B^2)

For the point C(0, k) and the line x + 7y = 11, we have:
Distance = |0 + 7k - 11| / √(1^2 + 7^2)
= |7k - 11| / √50

Calculating the Area:
Now that we have the base (√50) and the height (|7k - 11| / √50), we can substitute these values into the area formula.

Area = 1/2 * base * height
= 1/2 * √50 * (|7k - 11| / √50)
= 1/2 * |7k - 11|

Given that the area is 12 square units, we can write the equation:
1/2 * |7k - 11| = 12

Solving for k:
To solve for k, we need to consider two cases: when 7k - 11 is positive and when it is negative