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The diagram provided shows a task to construct a perpendicular bisector of line AB using a ruler and compass, with some points and distances already provided. Here's the breakdown of how to approach this problem:

1. **Drawing Line AB**:
   - Start by drawing a straight horizontal line AB with a length of 10 cm.

2. **Constructing the Perpendicular Bisector**:
   - Use a compass to find the perpendicular bisector. To do this:
     - Set the compass width to more than half the length of AB (e.g., slightly more than 5 cm).
     - Place the compass at point A and draw an arc above and below the line.
     - Without changing the compass width, place it at point B and draw another arc that intersects the previous ones.
     - The points where the arcs intersect above and below the line are equidistant from A and B. Draw a straight line through these intersection points. This is the perpendicular bisector of AB.

3. **Marking Point C**:
   - Point C is marked 2 cm above the perpendicular bisector. Use your ruler to measure 2 cm upward from the point where the bisector meets line AB and mark it as point C.

4. **Measuring AC**:
   - Use a ruler to measure the distance from point A to point C. Since the triangle formed is a right triangle, you can use the Pythagorean theorem to calculate the length of AC if needed:
     \[
     AC = \sqrt{\left(\frac{AB}{2}\right)^2 + (2\,\text{cm})^2} = \sqrt{5^2 + 2^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.39\,\text{cm}
     \]

Therefore, the length of \( AC \) is approximately **5.4 cm** (to 1 decimal place).

Would you like more details on any of the steps, or further clarification?

Here are five related questions to expand on this topic:
1. How would you find the midpoint of a line segment using only a compass and straightedge?
2. How can you construct a 90-degree angle using just a compass and straightedge?
3. What is the significance of a perpendicular bisector in geometry?
4. How does the Pythagorean theorem apply to this kind of construction?
5. How could you verify that your perpendicular bisector is accurate?

**Tip**: When constructing geometric shapes with a compass and straightedge, always make sure your compass is set accurately to avoid errors in your constructions.

Use a ruler and pair of compasses to make an accurate drawing of line AB and its perpendicular bisector. Mark point C on your drawing and measure the length of AC to 1 d.p.

Learn how to construct a perpendicular bisector using a ruler and compass and measure the length of a line segment in a geometry problem. This guide covers key geometry concepts such as perpendicular bisectors and the Pythagorean Theorem, suitable for students in Grades 8-10.

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