The problem in the image states:

"A rectangle ABCD has been divided into four smaller rectangles by drawing lines parallel to its sides. The perimeters of three of the smaller rectangles are 11, 16, and 19. Find the perimeter of the rectangle ABCD."

Solution Approach:

We are given three perimeters of smaller rectangles (11, 16, 19), and the goal is to find the perimeter of the original rectangle ABCD.

The perimeter of a rectangle is calculated by:

$P = 2(l + w)$

Where $l$ is the length and $w$ is the width of the rectangle.

Since ABCD has been divided into four smaller rectangles, each small rectangle has some combination of length and width from the original rectangle. If we consider two dimensions, the lengths and widths of these small rectangles, we can derive that they share these dimensions in common, which helps us solve for the total perimeter.

Steps:

1. Let’s denote the sides of the original rectangle as $L$ (length) and $W$ (width).

2. Each smaller rectangle's perimeter gives us information about how the sides are divided.

   • One small rectangle might have sides of $x$ and $y$,
   • Another might have $y$ and $z$,
   • And so on.

3. Summing the information from the small rectangles' perimeters, we get multiple equations involving the side lengths.

Let me know if you want a detailed step-by-step breakdown or if you would like help with the algebra.

Here are 5 related questions:

1. How do you calculate the area of the original rectangle ABCD?
2. If only two of the smaller rectangle perimeters were given, how would that change the problem?
3. What is the relationship between the side lengths of the smaller and larger rectangles?
4. How can you use perimeter formulas to reverse-engineer unknown dimensions?
5. Could a different number of rectangles change the final perimeter of the original rectangle?

Tip: Always remember the perimeter formula $P = 2(l + w)$ for a rectangle, and try to apply this principle to individual sections.