Final answer:
By equating the aspect ratio of the unfolded and folded paper, we determined that the length of the paper is also 21 cm, making the paper a square.
Step-by-step explanation:
To solve this problem, we must understand that when the sheet of paper is folded, the new width of the folded paper becomes half of the original width, and at the same time, the new length becomes the old width, while maintaining the shape. This implies that the aspect ratio before and after the fold is the same, because we are told the rectangles are "the same shape".
Let L be the length of the paper such that L/21 is the aspect ratio. After folding, the new dimensions are 21/2 cm by L cm, and the aspect ratio is L/(21/2) or 2L/21. Setting the aspect ratios equal because the shapes are the same, we have:
L/21 = 2L/21
Multiplying both sides by 21 to eliminate the denominator, we get:
L = 2L
Then divide both sides by L (assuming L is not zero) to find:
1 = 2
This equation has no solution, indicating that there was a mistake in our assumptions or method. The correct method would be to set the aspect ratios equal like so:
L/21 = 21/(L/2)
This gives us L^2 = 21 * 21
L^2 = 441
Therefore, L is the square root of 441, which is:
L = 21 cm
Hence, the length of the rectangular sheet of paper is 21 cm, amazingly the same as its width, indicating it is a square.