Using a ruler and compasses, which of the following constructions can be made?

Options

A. A circle, when the length of its radius is known.
B. The perpendicular bisector of a line segment of the given length.
C. Both A and B
D. None of these

Answer

To solve the question about which constructions can be made using a ruler and compass, we will analyze each option step by step.

### Step-by-Step Solution:

1. **Understanding Option A**:
- The option states that we can construct a circle when the length of its radius is known.
- **Construction Method**:
- Place the compass point on the desired center of the circle.
- Adjust the compass to the length of the radius.
- Rotate the compass 360 degrees to draw the circle.
- **Conclusion**: This construction is possible.

2. **Understanding Option B**:
- The option states that we can construct the perpendicular bisector of a line segment of a given length.
- **Construction Method**:
- Draw the line segment AB of the given length.
- With the compass, place the point on A and draw an arc above and below the line.
- Without changing the compass width, place the point on B and draw another arc that intersects the previous arcs.
- Label the intersection points as C and D.
- Draw a line through points C and D. This line is the perpendicular bisector of segment AB.
- **Conclusion**: This construction is also possible.

3. **Analyzing Option C**:
- Option C states that both A and B constructions can be made.
- Since both constructions are valid as established above, this option is correct.

4. **Analyzing Option D**:
- Option D states that none of the constructions can be made.
- Since we have established that both A and B can be constructed, this option is incorrect.

### Final Conclusion:
- The correct answer is **C**: both A and B constructions can be made using a ruler and compass.