https://file.aiquickdraw.com/mathbot/file/1724820488151ooipl0ew.jpg

To solve for the perimeter and area of the shape in the image, we'll break it down into manageable steps.

### Shape Breakdown:
The shape in the image appears to be a composite of rectangles. Let's analyze and break it down accordingly.

1. **Dimensions:**
   - The height of the top and bottom rectangles is labeled as \( x \).
   - The length of the vertical rectangle connecting the top and bottom rectangles is labeled as \( y \).
   - The total width of the shape (the width of the top and bottom rectangles) is labeled as \( z \).
   - The height of the central vertical section (which has no label) is \( y \).

### 1. **Perimeter Calculation:**

The perimeter is calculated by summing the lengths of all the outer sides of the shape.

- The top side's length is \( z \).
- The left vertical side's length is \( x + y + x = 2x + y \).
- The bottom side's length is \( z \) (since it's the same as the top).
- The right vertical side's length is also \( 2x + y \).

So, the total perimeter \( P \) is:
\[
P = 2z + 2(2x + y) = 2z + 4x + 2y
\]

### 2. **Area Calculation:**

To find the area, we can break the shape into three rectangles:
- The top rectangle has dimensions \( z \times x \), so its area is \( z \times x \).
- The bottom rectangle has the same dimensions \( z \times x \).
- The middle rectangle has dimensions \( (z - 2z) \times y \), which simplifies to \( z \times y \).

Thus, the total area \( A \) is:
\[
A = 2zx + zy
\]

### Summary:
- **Perimeter:** \( P = 2z + 4x + 2y \)
- **Area:** \( A = 2zx + zy \)

Would you like more details or have any questions about this solution?

Here are five related questions to deepen your understanding:
1. What happens to the perimeter if \( x \) is doubled?
2. How would the area change if \( y \) is halved?
3. Can this shape be divided in another way to simplify calculations?
4. What is the effect on the area and perimeter if all dimensions are scaled by a factor of 3?
5. How would you calculate the perimeter if the middle rectangle were not present?

**Tip:** When dealing with composite shapes, breaking them down into simpler shapes like rectangles or triangles can simplify calculations.

Learn how to calculate the perimeter and area of composite shapes using a detailed step-by-step guide. This problem involves identifying dimensions and applying formulas to find both perimeter and area. Suitable for high school students and beyond.

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