Final answer:
The validity of constructions made with a compass or straight edge is based on the foundational principles of geometry, which are a set of axioms and postulates that allow for logical deductions. Like in physics, where theories must align with nature and be experimentally verified, geometrical constructions align with these principles to ensure accurate outcomes. These methods, if applied correctly, provide consistent and reliable results.
Step-by-step explanation:
Knowing that constructions made with a compass or straight edge are valid relies on the principles of geometry. Geometry is established on a set of axioms and postulates that form the basis for logical deductions and constructions. For instance, we accept as a postulate that the angles in a triangle sum to 180 degrees, and we know three non-collinear points can define a plane. Constructions using a compass and straight edge are therefore valid because they abide by these fundamental truths. The fact remains consistent much like employing the Pythagorean Theorem, where if correctly applied, it provides a reliable result without fail. If we apply the same mathematical logic as in physics, where theories are verified by experiments and adherence to natural laws, we can trust the construction methods.
For example, when drawing a vector using a straight edge, you are translating the physical representation of a concept into a visual form. This helps to understand physical properties such as magnitude and direction. Even if we think about the input like a stone cut to fit precisely, we infer that there is an exact science behind it that involves the utilization of these tools and principles, which result in right angles or other geometric properties, even if in practice, like with the construction of a building, the angles may not be exactly 90 degrees.
Moreover, just as the needle of a compass aligns with a magnetic field when mapping its direction, the tools of geometry align with the axioms and postulates to ensure accurate and reproducible constructions. Thus, the trust in the validity of these geometrical constructions comes from their foundation on logical, mathematical principles and axioms that have been consistently verified as true.