To calculate the perimeter of the shaded part, we need to break the perimeter into three parts:

1. Arc of the larger sector (with radius $8 \, \text{cm}$),
2. Arc of the smaller sector (with radius $4.5 \, \text{cm}$),
3. The straight sides that join the two sectors (which are the radii of both sectors).

The formula for the length of an arc is given by:

$\text{Arc length} = \theta \times r$

where $\theta$ is the angle in radians and $r$ is the radius of the circle.

Since the given angle is $50^\circ$, we need to first convert this to radians:

$50^\circ = \frac{50 \times \pi}{180} \, \text{radians}$

Now, let's calculate the perimeter of the shaded part.

Steps:

1. Convert the angle $50^\circ$ to radians.
2. Calculate the arc length for the larger sector (radius $8 \, \text{cm}$).
3. Calculate the arc length for the smaller sector (radius $4.5 \, \text{cm}$).
4. Add the two arc lengths and the two radii.

I'll proceed with these calculations now.The perimeter of the shaded part is approximately 23.4 cm (rounded to 3 significant figures).

Would you like more details or have any questions?

Here are 5 related questions you might explore:

1. How is the arc length formula derived?
2. What happens to the perimeter if the angle increases?
3. How would the perimeter change if only the smaller radius changes?
4. Can this problem be generalized to other sector angles?
5. How would you calculate the area of the shaded part?

Tip: Always remember to convert angles from degrees to radians when dealing with arc length formulas.