To find the equation of the perpendicular bisector of the line segment whose endpoints are [tex]\((-1, 1)\)[/tex] and [tex]\( (7, -5)\)[/tex], we will follow these steps:
### Step 1: Find the Midpoint of the Line Segment
The midpoint of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) \][/tex]
For the endpoints [tex]\((-1, 1)\)[/tex] and [tex]\((7, -5)\)[/tex]:
[tex]\[ x_1 = -1, \, y_1 = 1, \, x_2 = 7, \, y_2 = -5 \][/tex]
So, the midpoint is:
[tex]\[ \left(\frac{-1 + 7}{2}, \frac{1 - 5}{2}\right) = \left(\frac{6}{2}, \frac{-4}{2}\right) = (3, -2) \][/tex]
### Step 2: Calculate the Slope of the Line Segment
The slope [tex]\( m \)[/tex] of a line segment with endpoints [tex]\((x_1, y_1)\)[/tex] and [tex]\( (x_2, y_2)\)[/tex] is given by:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
For the endpoints [tex]\((-1, 1)\)[/tex] and [tex]\((7, -5)\)[/tex]:
[tex]\[ m = \frac{-5 - 1}{7 - (-1)} = \frac{-6}{8} = -0.75 \][/tex]
### 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 segment. If the slope of the original line is [tex]\( m \)[/tex], the slope of the perpendicular bisector [tex]\( m' \)[/tex] is:
[tex]\[ m' = -\frac{1}{m} \][/tex]
For [tex]\( m = -0.75 \)[/tex]:
[tex]\[ m' = -\frac{1}{-0.75} = \frac{4}{3} \approx 1.3333 \][/tex]
### Step 4: Write the Equation in Point-Slope Form
The point-slope form of a line equation is:
[tex]\[ y - y_1 = m(x - x_1) \][/tex]
Here, we use the midpoint [tex]\( (3, -2) \)[/tex] as [tex]\((x_1, y_1)\)[/tex] and the slope [tex]\( m' = 1.3333 \)[/tex]:
[tex]\[ y - (-2) = \frac{4}{3}(x - 3) \][/tex]
Simplified, this becomes:
[tex]\[ y + 2 = \frac{4}{3}(x - 3) \][/tex]
This is the equation of the perpendicular bisector in point-slope form.
