To solve this problem, let's first analyze the relationship between the area and the perimeter of a square.

Step 1: Determine the side length of the original square.

Let the side length of the original square be $s$. The area of the square is given by:

$\text{Area} = s^2$

Step 2: Determine the new side length after halving the area.

If the area is halved, the new area is $\frac{s^2}{2}$. Let the new side length be $s'$. Then:

$(s')^2 = \frac{s^2}{2}$

Taking the square root of both sides:

$s' = \frac{s}{\sqrt{2}}$

Step 3: Calculate the perimeters.

• The perimeter of the original square is $4s$.
• The perimeter of the new square is $4s' = 4 \times \frac{s}{\sqrt{2}} = \frac{4s}{\sqrt{2}} = 2\sqrt{2}s$.

Step 4: Determine the decrease in perimeter.

The decrease in the perimeter is the difference between the original and the new perimeters:

$\text{Decrease in perimeter} = 4s - 2\sqrt{2}s = s(4 - 2\sqrt{2})$

Step 5: Simplify the expression.

The numerical value of $4 - 2\sqrt{2}$ is approximately:

$4 - 2\sqrt{2} \approx 4 - 2 \times 1.414 \approx 4 - 2.828 \approx 1.172$

So, the perimeter decreases by approximately $1.172 \times s$.

Conclusion:

The perimeter of the square would be decreased by approximately $s(4 - 2\sqrt{2})$, or roughly $1.172 \times s$, where $s$ is the original side length of the square.

Would you like more details or have any questions?

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1. How does the area of a square relate to its perimeter?
2. What is the exact value of $\sqrt{2}$?
3. How does changing the side length affect the perimeter of a square?
4. What is the relationship between area and side length in any polygon?
5. How would the answer change if the area was tripled instead of halved?

Tip: When dealing with square roots, it's often helpful to approximate their values to understand the magnitude of the result.