OBMEP Rectangular Sheet Problem: A Detailed Solution

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Hey guys! Today, we're diving deep into an intriguing problem from the Brazilian Mathematical Olympiad for Public Schools (OBMEP). This problem involves some rectangular sheets, a bit of cutting, and a whole lot of spatial reasoning. So, grab your thinking caps, and let's get started!

Understanding the Problem

Our main keyword here is the OBMEP rectangular sheet problem. The problem sets the stage with Lucinha, who has three identical rectangular sheets of paper. Each of these sheets measures 20 cm in width and 30 cm in length. Now, Lucinha takes the first sheet and makes two straight cuts. One cut is made 4 cm away from the left edge, and the other is made 7 cm away from the top edge. This effectively divides the original rectangle into four smaller rectangles. The challenge we face is to describe one of these resulting rectangles.

To truly understand this, imagine you have a physical sheet of paper in front of you. Picture making those cuts. The first cut, 4 cm from the left, creates a smaller rectangle on the left and a larger one on the right. The second cut, 7 cm from the top, further divides the sheet into four distinct rectangles. Each of these rectangles will have different dimensions, and that’s what we need to figure out. We need to visualize how the original dimensions of 20 cm and 30 cm are affected by these cuts. What are the lengths and widths of each of the four new rectangles? Are they all different, or are some of them the same? Thinking through these questions is key to solving the problem.

Moreover, consider the properties of rectangles themselves. Remember, a rectangle has four sides, with opposite sides being equal in length. All four angles in a rectangle are right angles (90 degrees). These basic properties will help us calculate the dimensions of the smaller rectangles. For instance, if we know one side of a rectangle and a cut is made parallel to that side, we can easily determine the new dimensions. By focusing on these fundamental geometric principles, we can approach the problem methodically and accurately. Remember, the beauty of geometry lies in its precision. Each line, each angle, and each dimension plays a crucial role in defining the shapes we encounter.

Visualizing the Cuts

To really nail this rectangular sheet problem, let's visualize those cuts step by step. Imagine the first cut, 4 cm from the left edge. This cut runs parallel to the longer side of the rectangle (30 cm). What does this tell us? Well, it creates two new rectangles. One rectangle has a width of 4 cm (the distance of the cut from the edge), and the other rectangle has a width that is the remainder of the original width (20 cm) after we subtract the 4 cm. That's 20 cm - 4 cm = 16 cm. So, we have rectangles with widths of 4 cm and 16 cm.

Now, let's think about the second cut, which is 7 cm from the top edge. This cut runs parallel to the shorter side of the original rectangle (20 cm). Similar to the first cut, this cut also creates two new rectangles. One rectangle has a length of 7 cm (the distance of this cut from the top), and the other rectangle’s length is what’s left of the original length (30 cm) after subtracting the 7 cm. That’s 30 cm - 7 cm = 23 cm. Now, we have rectangles with lengths of 7 cm and 23 cm.

Putting these cuts together, we can see how the original rectangle is divided into four smaller rectangles. Each of these rectangles is defined by a combination of the new widths (4 cm and 16 cm) and the new lengths (7 cm and 23 cm). Can you picture it? We have a small rectangle in the top-left corner, another in the top-right, one in the bottom-left, and the last one in the bottom-right. Each has unique dimensions based on the original cuts. Visualizing these rectangles is crucial for understanding the relationships between their sides and for calculating their areas, perimeters, or any other properties we might need to know.

To make it even clearer, you might want to sketch this out on a piece of paper. Draw a rectangle and then draw the two lines representing the cuts. Label the distances, and you’ll see the four rectangles taking shape. This hands-on approach can make the problem much more concrete and easier to solve. Remember, geometry is all about visualizing shapes and their properties. The more you practice visualizing, the better you'll become at solving these types of problems.

Describing the Rectangles

Okay, so we've made the cuts, and we've visualized the four rectangles. Now comes the key part of solving this OBMEP rectangular sheet problem: describing one of these rectangles. Remember, the cuts created rectangles with widths of 4 cm and 16 cm, and lengths of 7 cm and 23 cm. This gives us four possible rectangles:

  1. Rectangle 1: Width 4 cm, Length 7 cm
  2. Rectangle 2: Width 4 cm, Length 23 cm
  3. Rectangle 3: Width 16 cm, Length 7 cm
  4. Rectangle 4: Width 16 cm, Length 23 cm

Each of these rectangles is unique in its dimensions. Rectangle 1 is the smallest, with sides of 4 cm and 7 cm. Rectangle 4 is the largest, with sides of 16 cm and 23 cm. Rectangles 2 and 3 fall in between, each having one smaller side and one larger side. To fully describe each rectangle, we can state its dimensions: length and width. For example, we could say,