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To solve this problem, we need to determine the equation of the perpendicular bisector of the given line segment in slope-intercept form.

### Step 1: Analyze the given information
- The line segment shown on the graph has a midpoint at \( (3, 1) \).
- The line segment is drawn from point \( (2, 4) \) to \( (4, -2) \).

### Step 2: Find the slope of the original line segment
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

For the points \( (2, 4) \) and \( (4, -2) \):

\[
m = \frac{-2 - 4}{4 - 2} = \frac{-6}{2} = -3
\]

### Step 3: Find the slope of the perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of the original line. Therefore, if the slope of the original line is \( -3 \), the slope of the perpendicular bisector will be:

\[
m_{\text{perpendicular}} = \frac{1}{3}
\]

### Step 4: Use the point-slope form to find the equation
The point-slope form of the equation of a line is:

\[
y - y_1 = m(x - x_1)
\]

Using the midpoint \( (3, 1) \) and the slope \( \frac{1}{3} \):

\[
y - 1 = \frac{1}{3}(x - 3)
\]

### Step 5: Convert to slope-intercept form
Now, simplify the equation to get it in slope-intercept form \( y = mx + b \):

\[
y - 1 = \frac{1}{3}x - 1
\]
\[
y = \frac{1}{3}x - 1 + 1
\]
\[
y = \frac{1}{3}x
\]

### Conclusion:
The equation of the perpendicular bisector of the given line segment in slope-intercept form is:

\[
\boxed{y = \frac{1}{3}x}
\]

This corresponds to option A in the multiple-choice answers.

Would you like any further explanation on this problem? Here are some related questions:

1. How do you find the midpoint of a line segment?
2. What is the significance of the perpendicular bisector in geometry?
3. How can the slope of a line help in determining parallel and perpendicular lines?
4. What other forms can linear equations be expressed in?
5. Can a line segment have more than one perpendicular bisector?

**Tip:** Remember, when finding the perpendicular bisector, always use the negative reciprocal of the original line's slope.

Learn how to find the equation of the perpendicular bisector for the line segment connecting points (2, 4) and (4, -2) using geometry concepts and formulas. This problem involves calculating slopes, using point-slope and slope-intercept forms, and understanding the significance of perpendicular bisectors in geometry. Suitable for students in Grades 9-12.

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