Use the compass and ruler to construct the perpendicular bisector of the line segment \( \overline{A B} \).

Constructing the **perpendicular bisector** of a line segment \( \overline{AB} \) involves creating a line that is both perpendicular to \( \overline{AB} \) and divides it into two equal parts. Below are the step-by-step instructions to achieve this using only a compass and a ruler: ### Materials Needed: - A straightedge (ruler without markings) - A compass - Pencil - Paper ### Step-by-Step Guide: 1. **Draw the Line Segment \( \overline{AB} \):** - Begin by drawing a straight line segment and labeling its endpoints as \( A \) and \( B \). \[ A \longrightarrow \text{---} \longrightarrow B \] 2. **Set the Compass Width:** - Place the compass point on point \( A \). - Adjust the compass to a width greater than half the length of \( \overline{AB} \). This ensures that the arcs you draw will intersect. 3. **Draw Arcs Above and Below \( \overline{AB} \):** - With the compass set as above, draw an arc above the line. - Without changing the compass width, draw another arc below the line. These two arcs will intersect above and below the line segment. \[ \begin{align*} & \text{Set compass at } A: \quad \text{(Arc above)} \quad \text{(Arc below)} \\ & \text{Set compass at } B: \quad \text{(Arc above)} \quad \text{(Arc below)} \end{align*} \] 4. **Repeat the Arc Drawing from Point \( B \):** - Move the compass to point \( B \) while keeping the same width. - Draw another arc above \( \overline{AB} \), intersecting the first arc above. - Draw a second arc below \( \overline{AB} \), intersecting the first arc below. Now, you should have two intersection points: one above and one below \( \overline{AB} \). 5. **Mark the Intersection Points:** - Label the intersection point above \( \overline{AB} \) as \( C \). - Label the intersection point below \( \overline{AB} \) as \( D \). \[ C \quad \text{and} \quad D \] 6. **Draw the Perpendicular Bisector:** - Using the straightedge, draw a straight line passing through points \( C \) and \( D \). - This line intersects \( \overline{AB} \) at its midpoint and is perpendicular to \( \overline{AB} \). \[ \text{Perpendicular Bisector:} \quad CD \] 7. **Verify the Construction:** - **Perpendicular:** Measure the angle between \( \overline{AB} \) and the perpendicular bisector \( CD \) to ensure it is \( 90^\circ \). - **Bisector:** Measure the lengths \( \overline{AC} \) and \( \overline{CB} \) to confirm they are equal. ### Visual Representation: While a diagram would be ideal, here's a simplified textual representation: ``` C | |\ | \ A -----+----- B | / |/ D ``` - **\( AB \):** Original line segment. - **\( CD \):** Perpendicular bisector intersecting \( AB \) at its midpoint. ### Summary: By following these steps, you've successfully constructed the perpendicular bisector of \( \overline{AB} \). This bisector: - **Is perpendicular** to \( \overline{AB} \), forming a right angle. - **Bisects** \( \overline{AB} \) into two equal segments. This construction is fundamental in various geometric proofs and applications, ensuring precise and accurate results using only basic drawing tools.