To solve this problem, let's break down the information and reason through the dimensions and areas step by step.

Given:

1. A square is split vertically into left and right rectangles.
2. The right rectangle has an area of 7.
3. The left rectangle is split into a top square and a bottom rectangle, with the bottom rectangle having an area of 4.

Analysis:

Let the side length of the original square be $s$.

1. Right Rectangle:

   • The right rectangle has an area of 7.
   • If the width of the right rectangle is $w_r$, then the height of the rectangle must be $s$ (since it's part of the square).
   • Therefore, the area of the right rectangle can be expressed as: $s \times w_r = 7 \quad \text{(1)}$
   • This gives $w_r = \frac{7}{s}$.

2. Left Rectangle:

   • The left rectangle is split into a top square and a bottom rectangle.
   • Let the side length of the top square be $x$.
   • Since it's a square, its area will be $x^2$.
   • The bottom rectangle has an area of 4, and its height will be $s - x$ (because the total height of the square is $s$).

   Since the bottom rectangle is part of the left rectangle, its width is the same as the width of the left rectangle, denoted as $w_l$. The area of the bottom rectangle is given by: $w_l \times (s - x) = 4 \quad \text{(2)}$

3. Width Relationship:

   • The total width of the square is $s$, so: $w_l + w_r = s \quad \text{(3)}$

Substituting and Solving:

We have three key equations:

1. $s \times w_r = 7$
2. $w_l \times (s - x) = 4$
3. $w_l + w_r = s$

Let's solve for $s$, $w_l$, and $x$.

First, express $w_r$ from Equation (1): $w_r = \frac{7}{s}$ Now substitute into Equation (3): $w_l + \frac{7}{s} = s$ $w_l = s - \frac{7}{s} \quad \text{(4)}$

Substitute $w_l$ from Equation (4) into Equation (2): $\left(s - \frac{7}{s}\right) \times (s - x) = 4 \quad \text{(5)}$

Equation (5) is a nonlinear equation involving $s$ and $x$, which might require numerical or algebraic methods to solve explicitly. However, this equation essentially captures the relationship between the dimensions of the square and the areas given.

Conclusion:

The equations (4) and (5) describe the relationship between the original square's side length $s$, the width of the left rectangle $w_l$, and the top square's side length $x$. With this setup, you could solve for $s$ and $x$ numerically or algebraically to find the precise dimensions, depending on the exact values required.

Would you like to go further into solving these equations explicitly?

Follow-Up Questions:

1. How would the solution change if the right rectangle's area were different?
2. Can you find the exact value of $s$ if $x$ is known?
3. How does the side length $s$ relate to the overall structure if the top square's area is given?
4. What happens to the problem if the left rectangle's total area is specified instead of just the bottom rectangle?
5. What if the entire square’s side length were known, how would you solve for the dimensions of the rectangles?

Tip:

Always start with breaking down the problem into smaller, manageable parts, and use algebraic relationships to find connections between different elements of the problem.