Let's analyze the image and solve the given problem:

Part a: Find the area of the entire rectangle

To find the area of the entire rectangle, we need to multiply its length by its width.

From the diagram:

• The entire length of the rectangle is the sum of the two horizontal segments: $12 + 5 = 17$.
• The entire width of the rectangle is the sum of the two vertical segments: $6 + 4 = 10$.

Thus, the area $A$ of the rectangle is: $A = \text{length} \times \text{width} = 17 \times 10 = 170 \, \text{square units}.$

Part b: Calculate the perimeter of the figure

The perimeter of a rectangle is the sum of all its sides.

The formula for perimeter $P$ is: $P = 2 \times (\text{length} + \text{width})$ Using the total length $17$ and total width $10$: $P = 2 \times (17 + 10) = 2 \times 27 = 54 \, \text{units}.$

Would you like further explanation on how to approach this? Here are five related questions to practice with:

1. If the length of the rectangle is doubled, what would be the new area?
2. How would the perimeter change if the width is reduced by 3 units?
3. What is the diagonal length of the rectangle (use the Pythagorean theorem)?
4. Can you divide the same rectangle into different smaller rectangles and calculate their areas separately?
5. How would you find the perimeter if the rectangle was irregular?

Tip: When solving geometry problems involving areas and perimeters, always ensure the dimensions used are from the entire shape, not just parts.