The circumcentre of a triangle is determined by the :

Options

A. altitudes
B. median
C. perpendicular bisectors
D. angle bisectors

Answer

To solve the question regarding the circumcenter of a triangle, we will analyze the options provided and determine which one correctly identifies how the circumcenter is determined.

### Step-by-Step Solution:

1. **Understanding the Circumcenter**:
- The circumcenter of a triangle is the point where the perpendicular bisectors of the sides of the triangle intersect. It is equidistant from all three vertices of the triangle.

2. **Evaluating the Options**:
- **Option 1: Altitude** - The altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side. It does not determine the circumcenter.
- **Option 2: Median** - A median is a line segment joining a vertex to the midpoint of the opposite side. While medians are important in triangles, they do not determine the circumcenter.
- **Option 3: Perpendicular Bisector** - The perpendicular bisector of a side of a triangle is a line that is perpendicular to the side at its midpoint. The circumcenter is found at the intersection of the perpendicular bisectors of the triangle's sides.
- **Option 4: Angle Bisector** - An angle bisector divides an angle into two equal angles, but it does not determine the circumcenter.

3. **Conclusion**:
- The correct answer is **Option 3: Perpendicular Bisector**, as the circumcenter is determined by the intersection of the perpendicular bisectors of the sides of the triangle.

### Final Answer:
The circumcenter of a triangle is determined by the **Perpendicular Bisector**.

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