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.