Let's solve this problem step by step:

1. Understand the Problem:
   • Square $ABCD$ has side length 10.
   • $P$ is on the midpoint of side $AD$.
   • $Q$ is on the midpoint of side $BC$.
   • Connecting $P$ with $A$ and $C$, and $Q$ with $A$ and $C$, divides the square into three regions of equal area.
   • We need to find the length of segment $PQ$.

Step 1: Geometry and Coordinates

Let the square $ABCD$ be positioned in the coordinate plane as follows:

• $A(0, 0)$, $B(10, 0)$, $C(10, 10)$, $D(0, 10)$.

Midpoints:

• $P$ is the midpoint of $AD$: $P(0, 5)$.
• $Q$ is the midpoint of $BC$: $Q(10, 5)$.

Step 2: Analyze Area Conditions

The square is divided into three regions of equal area. The total area of the square is $100$, so each region must have an area of $\frac{100}{3}$.

Observations:

• The regions are formed by the diagonals $PA$, $PC$, $QA$, and $QC$.
• The diagonal lines divide the square into triangles and quadrilaterals, which balance the area constraint.

Step 3: Line Segment $PQ$

From the positions of $P(0, 5)$ and $Q(10, 5)$, it is clear that $PQ$ is a horizontal line segment because both $P$ and $Q$ lie on the same horizontal line $y = 5$.

Length of $PQ$:

The formula for the distance between two points is: $PQ = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}.$

Substitute: $PQ = \sqrt{(10 - 0)^2 + (5 - 5)^2} = \sqrt{10^2} = 10.$

Final Answer:

The length of $PQ$ is 10.

Would you like a detailed explanation of the area division or any other aspect? 😊

Here are 5 related questions you might find interesting:

1. How do we verify that the regions truly have equal areas?
2. What happens if $P$ and $Q$ are shifted off the midpoints?
3. Can you generalize this result for a square of side length $s$?
4. How does the geometry change if $P$ and $Q$ lie on different lines?
5. What is the significance of the horizontal alignment of $PQ$?

Tip: Always plot the points and check the symmetry in problems involving equal division of a figure.