The "perpendicular bisector" of the line segment \overline{AB}(The backslash then over line with the {AB} means that there's a line over AB) is the line that passes through the midpoint of \overline{AB} and is perpendicular to \overline{AB}.

To find the equation of the perpendicular bisector of a line segment \(\overline{AB}\), you need to follow these steps:

1. Determine the coordinates of points A and B. Let's assume A has coordinates \((x_1, y_1)\) and B has coordinates \((x_2, y_2)\).

2. Find the midpoint of the line segment \(\overline{AB}\) using the midpoint formula:
Midpoint coordinates = \(\left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)\).

3. Calculate the slope of the line segment \(\overline{AB}\) using the slope formula:
Slope of \(\overline{AB}\) = \(\frac{y_2 - y_1}{x_2 - x_1}\).

4. Determine the negative reciprocal of the slope found in step 3. Let's denote it as \(m\).

5. Apply the point-slope form of the equation of a line, using the midpoint coordinates from step 2 and the negative reciprocal slope from step 4:
Equation of the perpendicular bisector: \(y - y_{\text{midpoint}} = m(x - x_{\text{midpoint}})\).

6. Simplify the equation obtained in step 5 to any desired form, such as point-slope form, slope-intercept form, or standard form, if needed.

Following these steps will help you find the equation of the perpendicular bisector of the line segment \(\overline{AB}\).

To find the perpendicular bisector of the line segment AB, follow these steps:

1. Find the midpoint of AB: The midpoint of a line segment is the point that divides it into two equal parts. To find the midpoint of line segment AB, add the x-coordinates of A and B, and divide the sum by 2. Repeat this process for the y-coordinates. The resulting coordinates give you the midpoint.

2. Calculate the slope of AB: The slope of a line segment AB can be found using the formula (change in y-coordinates)/(change in x-coordinates), also known as (Δy)/(Δx). Subtract the y-coordinate of point A from that of point B, and divide it by the difference of their x-coordinates.

3. Find the negative reciprocal of AB's slope: The negative reciprocal of a slope is found by taking the negative of its reciprocal. In other words, you flip the fraction and change the sign. For example, if the slope of AB is m, then the negative reciprocal is -1/m.

4. Use the midpoint and negative reciprocal slope to form the equation: The equation of a line can be written in the form y = mx + b, where m is the slope and b is the y-intercept. Since you already have the slope (negative reciprocal of AB), substitute the midpoint coordinates into the equation and solve for b.

5. Write the equation of the perpendicular bisector: Now that you have the slope and y-intercept, substitute them into the equation y = mx + b. This will give you the equation of the line that represents the perpendicular bisector of AB.

By following these steps, you can find the equation of the perpendicular bisector of the line segment AB.