Find an equation for the perpendicular bisector of the line segment whose endpoints are left parenthesis, 5, comma, minus, 8, right parenthesis(5,−8) and left parenthesis, minus, 9, comma, minus, 4, right parenthesis(−9,−4).

To find the equation for the perpendicular bisector of a line segment, we need two main pieces of information: the midpoint of the line segment and the slope of the perpendicular bisector.

### Step 1: Find the Midpoint of the Line Segment
The midpoint \( M \) of a line segment with endpoints \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
For our endpoints \( (5, -8) \) and \( (-9, -4) \), we calculate:
\[
M = \left( \frac{5 + (-9)}{2}, \frac{-8 + (-4)}{2} \right) = \left( \frac{-4}{2}, \frac{-12}{2} \right) = (-2, -6)
\]

So, the midpoint is \( (-2, -6) \).

### Step 2: Find the Slope of the Line Segment
The slope \( m \) of a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Using our points \( (5, -8) \) and \( (-9, -4) \), we have:
\[
m = \frac{-4 - (-8)}{-9 - 5} = \frac{4}{-14} = -\frac{2}{7}
\]

### Step 3: Find the Slope of the Perpendicular Bisector
The perpendicular slope to \( m = -\frac{2}{7} \) is the negative reciprocal, so:
\[
m_{\text{perpendicular}} = \frac{7}{2}
\]

### Step 4: Write the Equation of the Perpendicular Bisector
Now we have the slope of the perpendicular bisector and a point on it, \( (-2, -6) \). Using the point-slope form of a line:
\[
y - y_1 = m (x - x_1)
\]
we substitute \( m = \frac{7}{2} \), \( x_1 = -2 \), and \( y_1 = -6 \):
\[
y - (-6) = \frac{7}{2}(x - (-2))
\]
Simplify:
\[
y + 6 = \frac{7}{2}(x + 2)
\]
Expanding this, we get:
\[
y + 6 = \frac{7}{2}x + 7
\]
Subtract 6 from both sides:
\[
y = \frac{7}{2}x + 1
\]

### Final Answer
The equation of the perpendicular bisector is:
\[
y = \frac{7}{2}x + 1
\]

Would you like more details on any part of this solution?

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Here are some related questions for further exploration:
1. How can we find the perpendicular bisector in three-dimensional space?
2. How can we determine if a point lies on a line given its equation?
3. How would we find the midpoint and perpendicular slope if one endpoint was unknown?
4. What if we want the equation in standard form instead of slope-intercept form?
5. How does the perpendicular bisector change if both endpoints have positive coordinates?

**Tip:** In geometry, perpendicular bisectors are useful for finding circumcenters in triangles, as the circumcenter lies at the intersection of the perpendicular bisectors of each side.

Find an equation for the perpendicular bisector of the line segment whose endpoints are (5,−8) and (−9,−4).

Discover how to determine the equation of the perpendicular bisector of a line segment given its endpoints, using midpoint, slope, and point-slope formulas. This detailed solution is ideal for students in Grades 9-12.

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