To determine the side length of the square $WXYZ$, we analyze the geometry in the figure step by step.

Observations:

1. $ABCD$ is a square with side length 1.
2. $P, Q, R, S$ are midpoints of sides $AB, BC, CD,$ and $DA$, respectively.
   • Hence, $AP = PB = 0.5$, $BQ = QC = 0.5$, $CR = RD = 0.5$, $DS = SA = 0.5$.
3. The vertices of square $WXYZ$ lie at intersections of lines connecting midpoints.

Step-by-Step Calculation:

1. Coordinates of the points

• Place $ABCD$ in a coordinate plane:
  • $A(0, 0), B(1, 0), C(1, 1), D(0, 1)$.
• Coordinates of midpoints:
  • $P(0.5, 0), Q(1, 0.5), R(0.5, 1), S(0, 0.5)$.

2. Equations of the lines through midpoints

• Line $PS$ passes through $P(0.5, 0)$ and $S(0, 0.5)$:
  • Slope $m = \frac{0.5 - 0}{0 - 0.5} = -1$.
  • Equation: $y = -x + 0.5$.
• Line $PR$ passes through $P(0.5, 0)$ and $R(0.5, 1)$:
  • Vertical line: $x = 0.5$.
• Line $QS$ passes through $Q(1, 0.5)$ and $S(0, 0.5)$:
  • Horizontal line: $y = 0.5$.
• Line $QR$ passes through $Q(1, 0.5)$ and $R(0.5, 1)$:
  • Slope $m = \frac{1 - 0.5}{0.5 - 1} = -1$.
  • Equation: $y = -x + 1.5$.

3. Intersection points (vertices of $WXYZ$)

• Solve for intersections of these lines:
  1. $PS: y = -x + 0.5$ and $QS: y = 0.5$:
     • Substitute $y = 0.5$: $0.5 = -x + 0.5$ $\Rightarrow x = 0$.
     • Intersection: $(0, 0.5)$.
  2. $PS: y = -x + 0.5$ and $PR: x = 0.5$:
     • Substitute $x = 0.5$: $y = -0.5 + 0.5 = 0$.
     • Intersection: $(0.5, 0)$.
  3. $QR: y = -x + 1.5$ and $QS: y = 0.5$:
     • Substitute $y = 0.5$: $0.5 = -x + 1.5$ $\Rightarrow x = 1$.
     • Intersection: $(1, 0.5)$.
  4. $QR: y = -x + 1.5$ and $PR: x = 0.5$:
     • Substitute $x = 0.5$: $y = -0.5 + 1.5 = 1$.
     • Intersection: $(0.5, 1)$.

Thus, vertices of $WXYZ$ are:

• $(0, 0.5), (0.5, 0), (1, 0.5), (0.5, 1)$.

4. Side length of $WXYZ$

• Distance between $(0, 0.5)$ and $(0.5, 0)$: $\text{Side length} = \sqrt{(0.5 - 0)^2 + (0 - 0.5)^2} = \sqrt{0.5^2 + 0.5^2} = \sqrt{0.25 + 0.25} = \sqrt{0.5}.$

Hence, the side length of square $WXYZ$ is $\sqrt{0.5} = \frac{\sqrt{2}}{2}$.

Let me know if you want further clarification or have additional questions!

Related Questions:

1. How would the calculation change if $ABCD$ had a side length other than 1?
2. Can you prove geometrically that $WXYZ$ is indeed a square?
3. How can the area of $WXYZ$ be calculated directly?
4. What is the ratio of the areas of $WXYZ$ to $ABCD$?
5. How would these results change if $P, Q, R,$ and $S$ were not midpoints?

Tip:

Whenever solving geometric problems involving coordinates, always identify key points and line equations carefully to simplify calculations.