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

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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.
To construct the perpendicular bisector of line segment $$ \overline{AB} $$ using a compass and ruler, follow these steps: 1. **Draw $$ \overline{AB} $$** on paper. 2. **Set the compass** to a width greater than half of $$ \overline{AB} $$. 3. **Draw arcs** above and below $$ \overline{AB} $$ from points $$ A $$ and $$ B $$. 4. **Mark the intersection points** $$ C $$ and $$ D $$ where the arcs cross. 5. **Draw the line** $$ CD $$ connecting $$ C $$ and $$ D $$. This line is the perpendicular bisector of $$ \overline{AB} $$. This bisector is perpendicular to $$ \overline{AB} $$ and divides it into two equal parts.

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

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