perpendicular bisector of a triangle
a line, ray, or segment that is perpendicular to a side of the triangle at the midpoint of the side
The perpendicular bisector of a triangle is a line or a segment that is perpendicular to one side of the triangle and passes through the midpoint of that side, dividing the side into two equal parts. Each side of a triangle has a perpendicular bisector, and they can intersect at a single point or be parallel.
The perpendicular bisectors of a triangle have several important properties. First, they are concurrent, meaning they all intersect at a single point called the circumcenter of the triangle. The circumcenter is equidistant from the vertices of the triangle and is the center of the circumcircle, the circle that passes through all three vertices of the triangle.
Second, the perpendicular bisector of a side is a line of symmetry for the triangle. This means that if a point is reflected across the perpendicular bisector of a side, it will be equidistant from the two endpoints of that side.
Finally, the length of the perpendicular bisector of a side is related to the lengths of the other sides of the triangle. Specifically, the length of the perpendicular bisector of a side equals half the square root of the sum of the squares of the other two sides. This is known as the perpendicular bisector theorem and can be used to find missing side lengths in a triangle.
Overall, the concept of the perpendicular bisector of a triangle is crucial in understanding the properties of triangles and is a fundamental component of geometry.
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