To construct the triangle $ABC$ and follow the steps described in the problem, we can break it down into a series of geometric constructions. I'll guide you through the process:

1. Construct Triangle $ABC$

• Given: $AB = 4 \, \text{cm}, BC = 8 \, \text{cm}, CA = 5.5 \, \text{cm}$.

• First, use a ruler and a compass to construct $AB$ with a length of 4 cm.

• Then, with a compass, place the point $A$ at one end of $AB$ and use the radius of 5.5 cm to draw an arc that intersects the circle.

• Similarly, with the point $B$ at the other end of $AB$, use a compass with a radius of 8 cm to draw another arc. This will define the position of point $C$.

• Mark the intersection of the two arcs as $C$. Connect $A$, $B$, and $C$ to form triangle $ABC$.

2. Construct a Circle with $AB$ as the Diameter

• With $AB$ as the diameter, set your compass to the length of $AB$ (4 cm).

• Place the compass at the midpoint of $AB$, which is 2 cm from both $A$ and $B$, and draw a circle with this radius.

3. Mark Point $P$ Equidistant from $AB$ and $AC$

• Point $P$ lies on the circumference of the circle. It should be equidistant from the line segment $AB$ and the line segment $AC$.

• To find point $P$, construct the angle bisector of $\angle ACB$ (the angle formed by $AC$ and $BC$). The angle bisector will divide the angle into two equal parts.

• Draw a line from $C$ that divides $\angle ACB$ into two equal angles. This bisector intersects the circle at some point, and this point is $P$, which will be equidistant from the lines $AB$ and $AC$.

4. Mark Point $Q$ Equidistant from $A$ and $B$

• To find point $Q$, construct the perpendicular bisector of the line segment $AB$.

• The perpendicular bisector will divide $AB$ into two equal segments. The point $Q$ should be on this line and equidistant from $A$ and $B$.

5. Measure Chord $PQ$

• Use a ruler to measure the length of the chord $PQ$ on the circle. This is the distance between points $P$ and $Q$.

By following the steps above, you'll be able to construct the required geometric configuration and measure the length of chord $PQ$.

Do you want further details or have any questions regarding the construction steps?

Here are five related questions you could explore further:

1. How do we calculate the length of a chord in a circle when given its distance from the center?
2. What is the significance of the perpendicular bisector in geometric constructions?
3. How can the concept of equidistant points be applied in other geometric figures?
4. Can we calculate the area of triangle $ABC$ using Heron’s formula?
5. What properties do angle bisectors and perpendicular bisectors have in a triangle?

Tip: When constructing geometric figures, it's always helpful to draw lightly first and then darken lines once you're sure about the placement of key points!