e the compass and ruler to construct the perpendicular bisector of the line segment $$ \overline{A B} $$.

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1. **Draw two circles:**  
   - Place the compass at point $$ A $$ and draw a circle with a radius greater than $$ \frac{AB}{2} $$.
   - Without changing the radius, place the compass at point $$ B $$ and draw another circle.  
   These two circles will intersect at two points.

2. **Identify intersection points:**  
   - Label the points where the circles intersect as $$ C $$ and $$ D $$.

3. **Draw the perpendicular bisector:**  
   - Use the ruler to draw the line through points $$ C $$ and $$ D $$.  
   This line is the perpendicular bisector of the line segment $$ \overline{AB} $$.

4. **Verify the construction:**  
   - The line $$ CD $$ will cut $$ \overline{AB} $$ into two equal parts.
   - Additionally, $$ CD $$ will be perpendicular to $$ \overline{AB} $$.

Thus, the perpendicular bisector of $$ \overline{AB} $$ has been constructed using only a compass and a ruler.
To construct the perpendicular bisector of $$ \overline{AB} $$ using a compass and ruler: 1. **Draw two circles** with centers at $$ A $$ and $$ B $$, each with a radius greater than half of $$ AB $$. 2. **Find the intersection points** of these circles, labeling them $$ C $$ and $$ D $$. 3. **Draw the line** through $$ C $$ and $$ D $$ using the ruler. This line is the perpendicular bisector of $$ \overline{AB} $$. 4. **Verify** that the line equally divides $$ \overline{AB} $$ and is perpendicular to it. This method ensures the accurate construction of the perpendicular bisector using only a compass and ruler.

e the compass and ruler to construct the perpendicular bisector of the line segment .

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