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Let's break down each of these scenarios to analyze their effect on the area and perimeter of a square.

1. **General Properties of a Square:**
   - Let the side of the square be \( s \).
   - The perimeter \( P \) of a square is given by \( P = 4s \).
   - The area \( A \) of a square is given by \( A = s^2 \).
   - The diagonal \( d \) of a square is given by \( d = s\sqrt{2} \).

Now let's consider each option:

### (a) Doubling the length of the sides:
- New side length = \( 2s \)
- **Perimeter**: \( P = 4 \times 2s = 8s \), so the perimeter is **doubled**.
- **Area**: \( A = (2s)^2 = 4s^2 \), so the area is **quadrupled**.

### (b) Halving the length of the sides:
- New side length = \( \frac{s}{2} \)
- **Perimeter**: \( P = 4 \times \frac{s}{2} = 2s \), so the perimeter is **halved**.
- **Area**: \( A = \left( \frac{s}{2} \right)^2 = \frac{s^2}{4} \), so the area is **reduced to a quarter**.

### (c) Tripling the length of the sides:
- New side length = \( 3s \)
- **Perimeter**: \( P = 4 \times 3s = 12s \), so the perimeter is **tripled**.
- **Area**: \( A = (3s)^2 = 9s^2 \), so the area is **multiplied by 9**.

### (d) Doubling the length of the diagonal distance:
- The diagonal of the square is \( d = s\sqrt{2} \). Doubling the diagonal gives \( 2d = 2s\sqrt{2} \).
- If the new diagonal is \( 2s\sqrt{2} \), the new side length is \( s\sqrt{2} \) (since \( s\sqrt{2} = \frac{d}{\sqrt{2}} \)).
- **Perimeter**: \( P = 4 \times s\sqrt{2} \).
- **Area**: \( A = (s\sqrt{2})^2 = 2s^2 \), so the area is **doubled**.

Would you like more details or have any questions? 

Here are 5 related questions:
1. How would increasing the side length by 50% affect the area and perimeter?
2. What happens to the area if the diagonal of the square is tripled?
3. How does the perimeter of a square compare to that of a circle with the same area?
4. What are the geometric properties of a rectangle compared to a square under these transformations?
5. How does scaling affect the volume in 3D shapes such as cubes?

**Tip:** The area of a shape is typically affected more drastically than its perimeter when dimensions are scaled.

What effect would each of the following have on the area and perimeter of a square?
(a) Doubling the length of the sides
(b) Halving the length of the sides
(c) Tripling the length of the sides
(d) Doubling the length of the diagonal distance between opposite corners

Explore how changing the side length or diagonal of a square affects its area and perimeter. Learn about the impact of doubling, halving, and tripling side lengths and how the diagonal distance influences the square's geometry. Suitable for students in Grades 7-10.

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