3.2.2 Using logic and sense-making, determine the perimeter of the combined shape. Justify your choice.

To accurately determine the perimeter of the combined shape and justify the choice, we need to consider the arrangement and the given dimensions carefully. Let's reassess the combined shape in the diagram.

### Given:
- The larger square has a side length of \( \sqrt{x} \).
- The smaller square has a side length of \( \sqrt{y} \).
- Dimensions: \( \sqrt{x} = 6 \) meters and \( \sqrt{y} = 4 \) meters.

### Arrangement:
The diagram shows two adjoining rooms where one side of each square aligns. This setup affects the perimeter because some sides are shared.

### Calculation of Combined Shape's Perimeter:

1. **Individual Perimeters:**
   - Larger square perimeter: \( 4 \times 6 = 24 \) meters.
   - Smaller square perimeter: \( 4 \times 4 = 16 \) meters.

2. **Combining the Shapes:**
   - When the squares are adjoining, they share a common side. This common side should not be counted twice in the combined perimeter.
   - The shared side length is 4 meters (side of the smaller square).

3. **Total Perimeter Calculation:**
   - Calculate the perimeter by adding the sides and subtracting the shared side.

\[ \text{Combined Perimeter} = (\text{Perimeter of larger square} + \text{Perimeter of smaller square}) - 2 \times (\text{shared side length}) \]
\[ \text{Combined Perimeter} = (24 + 16) - 2 \times 4 \]
\[ \text{Combined Perimeter} = 40 - 8 \]
\[ \text{Combined Perimeter} = 32 \text{ meters} \]

### Justification:
The correct perimeter of the combined shape is 32 meters. This is justified by:
- Adding the individual perimeters of both squares.
- Subtracting twice the shared side length to avoid over-counting.
- Considering the logical arrangement as shown in the diagram.

This approach ensures that all sides are counted accurately, reflecting the true combined perimeter without double-counting the shared boundary.

### Follow-up Questions and Tip:
Would you like more details or have any questions? Here are some more practice questions:
1. What is the combined area of the two squares?
2. How would the perimeter change if the shared side length were different?
3. How can we calculate the perimeter for shapes with more complex arrangements?
4. If the shapes were not adjoining, what would their total perimeter be?
5. How can understanding shared sides help in real-life construction projects?

**Tip:** When calculating the perimeter of combined shapes, always carefully consider shared sides to avoid double-counting and ensure accuracy.

Learn how to calculate the perimeter of combined shapes by considering shared sides. This problem involves calculating the perimeter of two adjoining squares with specific dimensions. Suitable for students in Grades 6-8.

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