The problem asks which of the listed constructions would help to create a line passing through point $C$ that is perpendicular to the line $AB$. Let's break down the options:

• Option A: Construction of an equilateral triangle with one side $AB$. This doesn't necessarily help create a perpendicular line through $C$, as it focuses on an equilateral triangle, which doesn't relate to perpendicularity.

• Option B: Construction of a hexagon with one side $BC$. This is also unrelated to constructing a perpendicular line through $C$.

• Option C: Construction of a perpendicular bisector through $C$. This is the correct option, as constructing a perpendicular bisector through point $C$ ensures that the line will intersect $AB$ at a 90-degree angle, which is exactly what the question asks for.

• Option D: Construction of a square with one side $AB$. Similar to option A, this focuses on creating a square and doesn't directly relate to the perpendicular line through $C$.

Thus, the correct answer is Option C.

Do you want more details or have any other questions?

Here are 5 related questions:

1. What is the definition of a perpendicular bisector?
2. How do you construct a perpendicular bisector using a compass?
3. Why does constructing a perpendicular bisector guarantee a 90-degree angle?
4. What are other methods to construct a perpendicular line through a point not on a line?
5. How do equilateral triangles and squares relate to perpendicularity in geometry?

Tip: When constructing a perpendicular bisector, ensure that both ends of the original line segment are equidistant from the point where the bisector intersects.