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Exploring the concept of perpendicular bisectors, this overview delves into their geometric properties, theorems like the Perpendicular Bisector Theorem, and points of concurrency such as the circumcenter and incenter in triangles. It also covers the centroid and orthocenter, highlighting their significance in triangle geometry and practical applications in solving geometric problems.
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A line or line segment that intersects another line segment at a 90-degree angle and cuts it into two congruent parts
Midpoint Formula
The formula used to find the midpoint of a line segment on a Cartesian plane
Relationship between slopes of perpendicular lines
The slopes of perpendicular lines are negative reciprocals of each other
A theorem stating that any point on the perpendicular bisector of a line segment is equidistant from the segment's endpoints
Points formed by the intersection of certain bisectors in a triangle
Circumcenter
The point where the perpendicular bisectors of a triangle's sides meet, equidistant from the triangle's vertices
Incenter
The point where the angle bisectors of a triangle's angles intersect, equidistant from the triangle's sides
Centroid
The point of intersection of the medians of a triangle, located at a distance of two-thirds from each vertex to the midpoint of the opposite side
Orthocenter
The point where the altitudes of a triangle intersect, with its position varying based on the type of triangle
The use of the Perpendicular Bisector Theorem and Angle Bisector Theorem to find unknown measurements in triangles
Understanding these theorems and their converses is crucial in comprehending the properties and structure of triangles