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The question asks to find the equation of the line \( CD \), which is the perpendicular bisector of the line segment \( AB \). Here's the process to solve it:

1. **Identify the coordinates of points \( A \) and \( B \)**: From the graph, point \( A \) appears to be at \( (4, 8) \) and point \( B \) at \( (8, 4) \).

2. **Find the midpoint of \( AB \)**:
   The midpoint \( M \) of \( AB \) is found using the midpoint formula:
   \[
   M = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)
   \]
   Substituting \( A(4, 8) \) and \( B(8, 4) \):
   \[
   M = \left( \frac{4 + 8}{2}, \frac{8 + 4}{2} \right) = (6, 6)
   \]

3. **Find the slope of \( AB \)**:
   The slope \( m_{AB} \) of line \( AB \) is calculated using the formula:
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   Substituting \( A(4, 8) \) and \( B(8, 4) \):
   \[
   m_{AB} = \frac{4 - 8}{8 - 4} = \frac{-4}{4} = -1
   \]

4. **Find the slope of the perpendicular bisector \( CD \)**:
   The slope of the perpendicular bisector is the negative reciprocal of the slope of \( AB \). Since the slope of \( AB \) is \( -1 \), the slope of \( CD \) will be:
   \[
   m_{CD} = 1
   \]

5. **Write the equation of the line \( CD \)**:
   The line \( CD \) passes through the midpoint \( M(6, 6) \) and has a slope of 1. The equation of a line is given by:
   \[
   y - y_1 = m(x - x_1)
   \]
   Substituting \( m = 1 \) and \( (x_1, y_1) = (6, 6) \):
   \[
   y - 6 = 1(x - 6)
   \]
   Simplifying:
   \[
   y = x
   \]

Thus, the equation of line \( CD \) is:
\[
y = x
\]

Do you have any questions or would you like further details on this?

Here are 5 related questions to explore:
1. How do you find the distance between two points on a coordinate plane?
2. What is the general method to find the equation of a perpendicular bisector?
3. How do you derive the slope-intercept form of a line from two points?
4. How can you verify that two lines are perpendicular?
5. What is the significance of the midpoint in geometry?

**Tip**: The slope of a line is crucial in many geometric problems—mastering its calculation helps in understanding parallelism and perpendicularity!

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?

Learn how to find the equation of the perpendicular bisector of a line segment on a coordinate plane. This problem involves using the midpoint formula, slope calculation, and line equations. Suitable for students in Grades 8-10.

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