Problem Breakdown:

We are given a compound L-shaped figure with known dimensions. The goal is:

1. Part (a): Find the unknown side length marked with ?.
2. Part (b): Calculate the perimeter of the shape after determining the missing length.

Step 1: Finding the unknown side length (a)

The compound shape is divided into two rectangles. By observing:

• The total vertical length of the left side is 12 cm.
• The top portion of the vertical length is 5 cm.
• Therefore, the bottom portion (the unknown length, ?) can be calculated as:

$? = 12 \, \text{cm} - 5 \, \text{cm} = 7 \, \text{cm}$

So, the missing side is: $\boxed{7 \, \text{cm}}$

Step 2: Calculating the Perimeter (b)

To calculate the perimeter, sum the lengths of all the sides of the compound shape. The dimensions are:

• Top horizontal: 5 cm
• Bottom horizontal: 9 cm
• Right vertical: 4 cm (top) + 4 cm (bottom) = 8 cm
• Left vertical: 12 cm

The perimeter is: $\text{Perimeter} = 5 \, \text{cm} + 9 \, \text{cm} + 4 \, \text{cm} + 4 \, \text{cm} + 12 \, \text{cm} + 7 \, \text{cm}$

Simplifying: $\text{Perimeter} = 41 \, \text{cm}$

So, the perimeter is: $\boxed{41 \, \text{cm}}$

Final Answer:

• (a): The unknown side length is 7 cm.
• (b): The perimeter of the shape is 41 cm.

Would you like a more detailed breakdown, or do you have further questions?
Here are five related questions to expand your understanding:

1. How would you calculate the area of this compound shape?
2. What is the difference between area and perimeter for compound shapes?
3. Can you find the perimeter if one of the side lengths is missing? Why or why not?
4. How would these calculations change if the figure was scaled by a factor of 2?
5. What units should be used for area compared to perimeter?

Tip: When working with compound shapes, break them into simpler parts (rectangles, triangles, etc.) for easier calculation!