A sheet of paper is rectangular, with a base equal to $20 \mathrm{~cm}$ and height $10 \mathrm{~cm}$. This sheet is folded along the dotted lines as shown in the figure below, and finally cut with scissors along the indicated line, which is parallel to the base and is at the midpoint of the triangle's height.

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-02.jpg?height=279&width=1513&top_left_y=1511&top_left_x=180)

a) After cutting at the indicated location, into how many parts is the sheet divided?

b) What is the area of the largest part?
Solution

a) Let's mark the line cut by the scissors in gray, and perform the reverse process, which corresponds to opening the sheet after it has been cut:

![](https://cdn.mathpix.com/cropped/2024_05_01_cc59d970f26105cc4ca1g-22.jpg?height=271&width=1557&top_left_y=984&top_left_x=158)

Therefore, the sheet was divided into three pieces.

b) As can be observed, the two squares cut out from the upper left and lower right corners have a side length of $5 \mathrm{~cm}$. Since the area of a square is the side length squared, the area of each square is $5 \times 5 = 25 \mathrm{~cm}^{2}$. The sheet is a rectangle with a base of $20 \mathrm{~cm}$ and a height of $10 \mathrm{~cm}$. Therefore, since the area of a rectangle is base times height, the area of the sheet is $20 \times 10 = 200 \mathrm{~cm}^{2}$. Subtracting the total area by the area of the two squares at the corners, we conclude that the area of the largest piece of the sheet after being cut by the scissors is

$$
200 - 2 \times 25 = 200 - 50 = 150 \mathrm{~cm}^{2}
$$