https://file.aiquickdraw.com/mathbot/file/1728670504354ng9gaywj.jpg

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

Which point could be on the line that is perpendicular to line MN and passes through point K?

Learn how to determine the coordinates of a point on a line that is perpendicular to another line and passes through a specified point. This problem explores key concepts in coordinate geometry, including slope calculations and the use of point-slope form. Suitable for students in Grades 8-10.

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