To construct the perpendicular bisector of a line segment, follow these steps:

1. Draw the Line Segment: Start by drawing the line segment $AB$ that you want to bisect.

2. Set Compass Width: Place the compass point on one endpoint of the line segment, $A$. Open the compass to slightly more than half the length of $AB$.

3. Draw Arcs from Endpoint $A$: With the compass set, draw an arc above and another arc below the line segment, keeping the compass point at $A$.

4. Draw Arcs from Endpoint $B$: Without changing the compass width, place the compass point on the other endpoint $B$. Draw another arc above and below the line segment to intersect the first set of arcs.

5. Mark Intersection Points: The two pairs of arcs should intersect above and below the line segment. Mark these intersection points as $P$ and $Q$.

6. Draw the Perpendicular Bisector: Use a straightedge to draw a line through points $P$ and $Q$. This line is the perpendicular bisector of segment $AB$. It bisects $AB$ at a 90° angle and passes through the midpoint of $AB$.

Now you have constructed the perpendicular bisector of $AB$!

Would you like a diagram or further clarification on any of these steps?

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Here are some follow-up questions to deepen your understanding:

1. How can you verify that the line is perpendicular to the segment?
2. Why does the perpendicular bisector pass through the midpoint?
3. Can you use this method for any line segment, regardless of its length?
4. What is the significance of a perpendicular bisector in geometry?
5. How would you find the coordinates of the midpoint if $A$ and $B$ have given coordinates?

Tip: When constructing arcs with a compass, ensure it is slightly more than half the segment to guarantee intersection points above and below the line.