Alright, let's start by analyzing the problem step-by-step.
Step 1: Perimeter of Square 1
Square 1 has a side length [tex]\( x = 2 \)[/tex].
- The perimeter [tex]\( P_1 \)[/tex] of a square is given by [tex]\( 4 \times \text{side length} \)[/tex].
[tex]\[
P_1 = 4 \times 2 = 8
\][/tex]
Step 2: Determining the Side Length of Square 2
Square 2 is formed by joining the midpoints of the sides of Square 1. When we join the midpoints of a square, the resulting shape is another square whose side is the length of the diagonal of the smaller squares formed by splitting Square 1 into four equal parts.
Visualize the smaller squares, each formed from half the sides of Square 1:
- The diagonal [tex]\( d \)[/tex] of each smaller square (which is the side length of Square 2) can be found using the Pythagorean theorem. Considering one of the smaller squares as having side [tex]\( \frac{x}{2} \)[/tex]:
[tex]\[
d = \sqrt{\left(\frac{x}{2}\right)^2 + \left(\frac{x}{2}\right)^2} = x \cdot \frac{1}{\sqrt{2}}
\][/tex]
With [tex]\( x = 2 \)[/tex]:
[tex]\[
d = 2 \cdot \frac{1}{\sqrt{2}} = \frac{2}{\sqrt{2}} = \sqrt{2}
\][/tex]
So, the side length [tex]\( y \)[/tex] of Square 2 is [tex]\( \sqrt{2} \)[/tex].
Step 3: Perimeter of Square 2
- The perimeter [tex]\( P_2 \)[/tex] of Square 2 is given by [tex]\( 4 \times \text{side length} \)[/tex].
[tex]\[
P_2 = 4 \times \sqrt{2} \approx 5.65685424949238
\][/tex]
Step 4: Ratio of the Perimeters of Square 1 to Square 2
Now, we need to find the ratio of [tex]\( P_1 \)[/tex] to [tex]\( P_2 \)[/tex]:
[tex]\[
\text{Ratio} = \frac{P_1}{P_2} = \frac{8}{4\sqrt{2}} = \frac{8}{4 \times 1.4142135623730951} = \frac{8}{5.65685424949238} \approx 1.4142135623730951 \approx \sqrt{2}
\][/tex]
Thus, the ratio of the perimeter of Square 1 to the perimeter of Square 2 simplifies to [tex]\( \sqrt{2} \)[/tex], which corresponds to [tex]\(\boxed{1: \sqrt{2}}\)[/tex].
So, the correct answer is:
C. [tex]\(1: \sqrt{2}\)[/tex]
