https://file.aiquickdraw.com/mathbot/file/172696726061856b1a5sr.jpg

Let's analyze the image and solve the problem step by step.

### Problem Breakdown:

1. **Line segment \( AB \)** is shown on the coordinate plane.
2. We are tasked with finding the equation of line \( CD \), which is the perpendicular bisector of \( AB \).
3. A perpendicular bisector of a line segment:
   - Is perpendicular to the line.
   - Intersects the line at its midpoint.

### Step 1: Coordinates of points A and B
From the graph:
- Point \( A \) is at \( (4, 6) \).
- Point \( B \) is at \( (12, 2) \).

### Step 2: Midpoint of line segment \( AB \)
The midpoint formula is:
\[
M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
\]
Using \( A(4, 6) \) and \( B(12, 2) \):
\[
M = \left( \frac{4 + 12}{2}, \frac{6 + 2}{2} \right) = (8, 4)
\]

### Step 3: Slope of line segment \( AB \)
The slope formula is:
\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]
Substitute the coordinates of \( A(4, 6) \) and \( B(12, 2) \):
\[
m_{AB} = \frac{2 - 6}{12 - 4} = \frac{-4}{8} = -\frac{1}{2}
\]

### Step 4: Slope of perpendicular bisector
The slope of the perpendicular bisector is the negative reciprocal of the slope of \( AB \). So, if the slope of \( AB \) is \( -\frac{1}{2} \), the slope of \( CD \) will be:
\[
m_{CD} = 2
\]

### Step 5: Equation of line \( CD \)
The equation of a line in slope-intercept form is:
\[
y - y_1 = m(x - x_1)
\]
Using the midpoint \( M(8, 4) \) and the slope \( 2 \), we can plug into the point-slope form:
\[
y - 4 = 2(x - 8)
\]
Simplifying:
\[
y - 4 = 2x - 16
\]
\[
y = 2x - 12
\]

### Final Answer:
The equation of line \( CD \) is:
\[
y = 2x - 12
\]

Would you like more details or explanations on any part? 

Here are some related questions:
1. How do you find the slope of a line between two points?
2. What is the significance of the perpendicular bisector in geometry?
3. How do you use the midpoint formula to find the center of a line segment?
4. What are the characteristics of parallel and perpendicular lines?
5. How can we check if two lines are perpendicular using their slopes?

**Tip:** The negative reciprocal rule is key when working with perpendicular lines—always remember to flip and change the sign of the slope.

Line segment AB is shown on the coordinate plane. Line CD is the perpendicular bisector of line AB. A perpendicular bisector is perpendicular to the given line segment and intersects the line segment at its midpoint. What is the equation of line CD?

Solve a geometry problem involving the perpendicular bisector of line segment AB. This solution explains how to find the midpoint, calculate slopes, and use the point-slope formula to determine the equation of line CD. Suitable for Grades 9-11.

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