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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.

Which of the following constructions would help to construct a line passing through point C that is perpendicular to the line AB?

Learn how to construct a line passing through point C that is perpendicular to line AB using the perpendicular bisector method. This geometry problem involves understanding perpendicularity and geometric constructions, suitable for students in Grades 9-10.

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