To solve this accurately, let’s break it into steps based on the geometry construction process:

1. Draw Line AB:

   • Using a ruler, draw a horizontal line $AB$ that is 10 cm long.

2. Construct the Perpendicular Bisector:

   • Using a compass, set the radius to more than half the length of $AB$ (e.g., 6 cm).
   • Place the compass point on $A$ and draw arcs above and below the line.
   • Without changing the compass width, place the point on $B$ and draw similar arcs that intersect the previous arcs.
   • Mark the points where the arcs intersect above and below $AB$. Connect these points using a ruler to create the perpendicular bisector.

3. Locate the Midpoint and Point C:

   • The perpendicular bisector will cross $AB$ at its midpoint (5 cm from $A$).
   • Measure 2 cm vertically upwards from this midpoint along the perpendicular bisector to mark point $C$.

4. Measure AC:

   • $AC$ is the hypotenuse of a right triangle with the following:
     • One leg = 5 cm (half of $AB$).
     • The other leg = 2 cm (vertical distance to $C$).
   • Use the Pythagorean theorem: $AC = \sqrt{(5)^2 + (2)^2} = \sqrt{25 + 4} = \sqrt{29} \approx 5.4 \, \text{cm (to 1 decimal place)}.$

Solution: The length of $AC$ is 5.4 cm.

Let me know if you would like detailed diagrams or further clarifications.

Questions:

1. What are the general steps to construct a perpendicular bisector?
2. How does the Pythagorean theorem apply to such geometric problems?
3. What tools ensure maximum accuracy in compass constructions?
4. Why is it important to extend arcs when drawing bisectors?
5. How can you verify the accuracy of your perpendicular bisector?

Tip: Always double-check the compass settings when drawing arcs to ensure precision.