https://file.aiquickdraw.com/mathbot/file/1727226808930k6ne55v7.jpg

To find the equation of the perpendicular bisector of the segment joining the points \((8, -6)\) and \((-4, 5)\), we can break the process into two steps:

### Step 1: Find the midpoint of the segment
The midpoint of a segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[
\text{Midpoint} = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]

Substitute the given points:
\[
\text{Midpoint} = \left( \frac{8 + (-4)}{2}, \frac{-6 + 5}{2} \right) = \left( \frac{4}{2}, \frac{-1}{2} \right) = (2, -0.5)
\]
So, the midpoint is \((2, -0.5)\).

### Step 2: Find the slope of the perpendicular bisector
The slope of the line joining two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{Slope of the line segment} = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substitute the points \((8, -6)\) and \((-4, 5)\):
\[
\text{Slope of the line segment} = \frac{5 - (-6)}{-4 - 8} = \frac{11}{-12} = -\frac{11}{12}
\]

The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment:
\[
\text{Slope of the perpendicular bisector} = \frac{12}{11}
\]

### Step 3: Use the point-slope form of the line equation
The equation of a line with slope \(m\) passing through a point \((x_1, y_1)\) is given by:
\[
y - y_1 = m(x - x_1)
\]

Substitute the slope \( \frac{12}{11} \) and the midpoint \((2, -0.5)\):
\[
y - (-0.5) = \frac{12}{11}(x - 2)
\]
Simplifying:
\[
y + 0.5 = \frac{12}{11}(x - 2)
\]
Multiply out:
\[
y + 0.5 = \frac{12}{11}x - \frac{24}{11}
\]
Subtract 0.5 (or \(\frac{11}{22}\)) from both sides:
\[
y = \frac{12}{11}x - \frac{24}{11} - \frac{11}{22}
\]
\[
y = \frac{12}{11}x - \frac{35}{22}
\]

Thus, the equation of the perpendicular bisector is:
\[
y = \frac{12}{11}x - \frac{35}{22}
\]

Would you like more details or have any further questions?

Here are 5 related questions:
1. How do you find the slope between any two points?
2. What is the geometric significance of a perpendicular bisector?
3. Can the equation of the perpendicular bisector change if the line segment is rotated?
4. How does finding the midpoint help in solving this kind of problem?
5. Can you find the perpendicular bisector in 3D space?

**Tip:** Always remember that the perpendicular bisector has a slope that is the negative reciprocal of the original line's slope.

Find the equation of the perpendicular bisector of the segment joining the points (8, -6) and (-4, 5).

Learn how to find the equation of the perpendicular bisector for the line segment joining points (8, -6) and (-4, 5). This solution involves finding the midpoint and using slope formulas. Suitable for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation