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.