Let's solve the problem based on the information in the image.

Step 1: Find the slope of line $\overline{MN}$

To find the slope of line $\overline{MN}$, we can use the coordinates of points $M$ and $N$.

• Coordinates of $M$ (from the graph): $M(2, 4)$
• Coordinates of $N$ (from the graph): $N(-3, 2)$

The formula for the slope between two points $(x_1, y_1)$ and $(x_2, y_2)$ is:

$\text{slope of } \overline{MN} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - 2}{2 - (-3)} = \frac{2}{5}$

Step 2: Find the slope of the line perpendicular to $\overline{MN}$

The slope of the line perpendicular to $\overline{MN}$ is the negative reciprocal of $\frac{2}{5}$. So:

$\text{slope of perpendicular line} = -\frac{5}{2}$

Step 3: Equation of the line passing through point $K(0, -4)$

Now, we need to find the equation of the line that is perpendicular to $\overline{MN}$ and passes through point $K(0, -4)$. The point-slope form of the equation of a line is:

$y - y_1 = m(x - x_1)$

Substituting $m = -\frac{5}{2}$, and $K(0, -4)$:

$y - (-4) = -\frac{5}{2}(x - 0)$

Simplifying:

$y + 4 = -\frac{5}{2}x$

$y = -\frac{5}{2}x - 4$

Step 4: Check which point lies on this line

We now check the given options by substituting the coordinates into the equation $y = -\frac{5}{2}x - 4$.

1. For $(0, -12)$: $y = -\frac{5}{2}(0) - 4 = -4 \quad (\text{not } -12)$

2. For $(2, 2)$: $y = -\frac{5}{2}(2) - 4 = -5 - 4 = -9 \quad (\text{not } 2)$

3. For $(4, 8)$: $y = -\frac{5}{2}(4) - 4 = -10 - 4 = -14 \quad (\text{not } 8)$

4. For $(5, 13)$: $y = -\frac{5}{2}(5) - 4 = -\frac{25}{2} - 4 = -\frac{25}{2} - \frac{8}{2} = -\frac{33}{2} = -16.5 \quad (\text{not } 13)$

None of these points satisfy the equation, but this is an issue with matching the graph correctly based on how numbers fit. So I