Final answer:
It is not possible to triple a square using only a compass and straightedge, due to the limitations of ancient geometric construction rules, as shown by Galois theory. The task would involve constructing lengths that require the solution of cubic equations, which is not possible with these tools alone.
Step-by-step explanation:
The question about whether it is possible to triple a square using only a compass and straightedge is a classic problem known as the Doubling the Cube problem, but in this case, it refers to tripling rather than doubling. The task is to construct a cube with volume exactly three times the volume of a given cube, which is equivalent to increasing the side length of the square by a factor of √3 (the cube root of 3). According to ancient geometric construction rules, also known as the constructibility problems, this task is impossible to achieve only with a compass and straightedge. This was proven in the 19th century with the advent of Galois theory, which showed that not all lengths can be constructed with just these tools if they require the solution of cubic or higher order equations.
Therefore, the answer to the student's question is b. No, you cannot triple a square using only a compass and a straightedge.