Perpendicular Bisectors & Circumcenter
The perpendicular bisector of a segment is the set of all points equidistant from both endpoints. In a triangle, the three perpendicular bisectors meet at a single point — the circumcenter.
Introduction
The perpendicular bisector of a segment cuts it in half at a right angle. The key property: any point on the perpendicular bisector is equidistant from both endpoints. When applied to all three sides of a triangle, this gives us the circumcenter — the center of the circle that passes through all three vertices (the circumscribed circle).
Past Knowledge
Midpoints (1.2). Perpendicular lines (3.1). Midsegments (6.1.1).
Today's Goal
Use the Perpendicular Bisector Theorem and locate the circumcenter.
Future Success
Angle bisectors & incenter (6.1.3), circle theorems (Ch 10).
Key Concepts
Perpendicular Bisector Theorem
If a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. Conversely, if a point is equidistant from two endpoints, it lies on their perpendicular bisector.
Circumcenter
The circumcenter is the point where all three perpendicular bisectors of a triangle's sides meet.
- It is equidistant from all three vertices
- It is the center of the circumscribed circle (the circle through all three vertices)
- For acute triangles: inside. For right triangles: on the hypotenuse. For obtuse triangles: outside.
Worked Examples
Using the Perp. Bisector Theorem
Point lies on the perpendicular bisector of . If , find .
By the Perpendicular Bisector Theorem, is equidistant from and .
Finding the Circumradius
The circumcenter of is at point . If , find and .
The circumcenter is equidistant from all three vertices.
(circumradius = 12)
Coordinate Circumcenter
Find the circumcenter of the triangle with vertices , , .
This is a right triangle (right angle at ).
Circumcenter of a right triangle is the midpoint of the hypotenuse.
Hypotenuse : midpoint =
Circumcenter =
Common Pitfalls
Thinking the Circumcenter Is Always Inside
For obtuse triangles, the circumcenter is outside the triangle. For right triangles, it's on the hypotenuse.
Confusing Circumcenter with Centroid
The circumcenter uses perpendicular bisectors. The centroid uses medians. They are different points (except in equilateral triangles).
Real-Life Applications
Cell Tower Placement
When placing a cell tower equidistant from three towns, engineers find the circumcenter of the triangle formed by the three locations — ensuring equal signal coverage.
Emergency Services
Fire stations and hospitals are sometimes located at circumcenter positions to ensure equal response times to surrounding communities.
Practice Quiz
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