The problem provides a large rectangle divided into four smaller rectangles, three of which have known perimeters, and one whose perimeter we need to find.

Step 1: Define the dimensions of the large rectangle

From the diagram, we can break down the large rectangle based on the divisions made by the smaller rectangles. The following variables represent the dimensions:

• Let the height of the large rectangle be $h = 7$ (units).
• Let the width of the large rectangle be $w = 16$ (units).

Step 2: Assign variables to the unknown dimensions

We can divide the width and height into segments:

• Let $x$ be the width of the leftmost small rectangle.
• Let $y$ be the height of the top-left small rectangle.

Thus:

• The height of the two bottom rectangles is $7 - y$.
• The width of the two right rectangles is $16 - x$.

Step 3: Use the given perimeters

We can use the perimeters of the small rectangles to set up equations for their dimensions. The perimeter $P$ of a rectangle is given by the formula: $P = 2 \times (\text{length} + \text{width}).$

Perimeter 1: $10$ units (top-left rectangle)

$2(x + y) = 10 \quad \Rightarrow \quad x + y = 5.$

Perimeter 2: $6$ units (bottom-left rectangle)

The height of this rectangle is $7 - y$, and its width is $x$. $2(x + (7 - y)) = 6 \quad \Rightarrow \quad x + (7 - y) = 3 \quad \Rightarrow \quad x - y = -4.$

Perimeter 3: $25$ units (bottom-right rectangle)

The height of this rectangle is $7 - y$, and its width is $16 - x$. $2((16 - x) + (7 - y)) = 25 \quad \Rightarrow \quad (16 - x) + (7 - y) = 12.5.$

Step 4: Solve the system of equations

We have two key equations from the first two perimeters:

1. $x + y = 5$,
2. $x - y = -4$.

Add these two equations: $(x + y) + (x - y) = 5 + (-4) \quad \Rightarrow \quad 2x = 1 \quad \Rightarrow \quad x = 0.5.$

Substitute $x = 0.5$ into $x + y = 5$: $0.5 + y = 5 \quad \Rightarrow \quad y = 4.5.$

Step 5: Use the dimensions to find the unknown perimeter

The top-right rectangle has dimensions:

• Width: $16 - x = 16 - 0.5 = 15.5$,
• Height: $y = 4.5$.

Now, calculate the perimeter: $P = 2 \times (15.5 + 4.5) = 2 \times 20 = 40.$

Thus, the perimeter of the fourth rectangle is $\boxed{40}$.

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Would you like to go over these steps in more detail, or explore other related problems?

Here are 5 related questions for further exploration:

1. How does changing the height of the large rectangle affect the solution?
2. Can you find the area of each small rectangle?
3. How would the problem change if the given perimeters were areas instead?
4. What if only two of the smaller rectangle perimeters were known?
5. How can this approach be applied to irregular shapes?

Tip: When solving geometry problems involving rectangles, it's often helpful to label all dimensions and set up equations early in the process.