Certainly! Let's work through the problem step-by-step:
### Step 1: Identify Given Information
We are given:
1. The midpoint of the line segment is at [tex]\( (3, 1) \)[/tex].
2. The equation of the line segment is [tex]\( y = \frac{1}{3} x \)[/tex].
### Step 2: Determine the Perpendicular Slope
The slope of the given line [tex]\( y = \frac{1}{3} x \)[/tex] is [tex]\( \frac{1}{3} \)[/tex].
For the perpendicular bisector, the slope will be the negative reciprocal of [tex]\( \frac{1}{3} \)[/tex]. The negative reciprocal is calculated as:
[tex]\[ \text{Perpendicular Slope} = -\frac{1}{\frac{1}{3}} = -3 \][/tex]
### Step 3: Use the Perpendicular Slope and Midpoint to Find the Y-Intercept
The equation of a line in slope-intercept form is [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
Let's use the point-slope form of the equation to find [tex]\( b \)[/tex]:
[tex]\[ y = mx + b \][/tex]
Plug in the midpoint [tex]\( (3, 1) \)[/tex] and the perpendicular slope [tex]\( -3 \)[/tex]:
[tex]\[ 1 = -3 \cdot 3 + b \][/tex]
[tex]\[ 1 = -9 + b \][/tex]
[tex]\[ b = 1 + 9 \][/tex]
[tex]\[ b = 10 \][/tex]
### Step 4: Formulate the Equation of the Perpendicular Bisector
Putting it all together, we have:
[tex]\[ \text{Slope} = -3 \][/tex]
[tex]\[ \text{Y-Intercept} = 10 \][/tex]
So, the equation of the perpendicular bisector in slope-intercept form is:
[tex]\[ y = -3x + 10 \][/tex]
### Step 5: Match the Equation with the Given Options
From the given options:
- [tex]\( y = \frac{1}{3} x \)[/tex]
- [tex]\( y = \frac{1}{3} x - 2 \)[/tex]
- [tex]\( y = 3 x \)[/tex]
- [tex]\( y = 3 x - 8 \)[/tex]
None of these match the equation we have found ([tex]\( y = -3x + 10 \)[/tex]) exactly.
Thus, the valid equation for the perpendicular bisector is not among the given options.
