How to Find the Circumcenter of a Triangle Using Perpendicular Bisectors: A Step-by-Step Guide

circumcenter

The circumcenter is a point that is equidistant from the vertices of a triangle

The circumcenter is a point that is equidistant from the vertices of a triangle. In other words, it is the center of the unique circle that passes through all three vertices of the triangle.

To find the circumcenter of a triangle, you can use a few different methods. Here, I will explain the most common one using the perpendicular bisectors of the sides.

1. Start by drawing a triangle with three labeled vertices: A, B, and C.

2. Take any two sides of the triangle (say, AB and BC) and find the midpoint of each side. Let’s call these midpoints M (midpoint of AB) and N (midpoint of BC).

3. Determine the slopes of the two sides AB and BC. The slope of a line passing through two points (x1, y1) and (x2, y2) is given by (y2 – y1)/(x2 – x1).

4. Find the negative reciprocal of the slope of AB. This will be the slope of the perpendicular bisector of AB. Let’s call this slope m1.

5. Similarly, find the negative reciprocal of the slope of BC. Let’s call this slope m2.

6. Using the point-slope form of a line (y – y1) = m(x – x1), where (x1, y1) is the midpoint of the side and m is the slope, write the equation for the line passing through M with slope m1 and the equation for the line passing through N with slope m2.

7. Solve the two equations simultaneously to find the point of intersection. This point represents the circumcenter of the triangle.

8. Finally, confirm that the distance from this point to each of the three vertices (A, B, and C) is the same. You can use the distance formula, which is given by the square root of [(x2 – x1)^2 + (y2 – y1)^2] for two points (x1, y1) and (x2, y2).

By following these steps, you can find the circumcenter of any given triangle. Remember to be careful with your calculations and check your work to ensure accuracy.

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