How to Find the Circumcenter of a Triangle | Step-by-Step Guide and Applications

Circumcenter (defn)

The circumcenter of a triangle is the point at which the perpendicular bisectors of the triangle’s sides intersect

The circumcenter of a triangle is the point at which the perpendicular bisectors of the triangle’s sides intersect. In simpler terms, it is the center of a circle that passes through all three vertices of the triangle.

To find the circumcenter of a triangle, you can follow these steps:

1. Take any two sides of the triangle and find their midpoints.
– To find the midpoint, add the two coordinates of each side together and divide by 2.

2. Find the slopes of the lines that are perpendicular to the given sides.
– The slope of a line perpendicular to another line is the negative reciprocal of its slope. So, if one side has a slope of m, the perpendicular slope will be -1/m.

3. Use the midpoint of each side and its perpendicular slope to build the equations of the perpendicular bisectors.
– Use the point-slope form (y – y1) = m(x – x1), where (x1, y1) is the midpoint and m is the slope.

4. Solve the system of equations formed by the two equations of the perpendicular bisectors to find their point of intersection.
– Set the two equations equal to each other and solve for x and y.

The resulting x and y values represent the coordinates of the circumcenter of the triangle.

Finding the circumcenter is useful in geometry and trigonometry because it helps determine properties of triangles. For example, the distance between the circumcenter and any vertex of the triangle is the same. This distance is called the circumradius, and it is useful in calculations involving inscribed circles and orthocenters of triangles.

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