Final answer:
The answer to whether you can triple the area of a square using only a compass and straightedge is 'No.' This is because it involves constructing lengths that are not constructible with these tools only, similar to the classical problem of doubling a square.
Step-by-step explanation:
The question is asking whether it is possible to triple the area of a square using only a compass and straightedge, which is a classical problem in geometry known as the doubling the square problem. Unfortunately, the answer is 'No.' While the ancient Greeks gave us a wealth of knowledge in geometry, they also had problems they could not solve, known as the three famous problems of antiquity. One of these problems is to double the area of a square, also known as the Delian problem. Doubling a square is equivalent to constructing a square with twice the area of a given one, using just a compass and straightedge. This is an impossible task because it involves constructing a length of √2 times the side of the original square, which cannot be done with just those tools, since √2 is a non-constructible number. The problem of tripling a square's area is equally impossible for the same reasons; it would require constructing a length √3 times the side of the original square.