Feedback

What do you think about us?

Your name

Your email

Message

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.

Show More

## Perpendicular Bisectors

### Definition

A line or line segment that intersects another line segment at a 90-degree angle and cuts it into two congruent parts

### Properties

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

### The Perpendicular Bisector Theorem

A theorem stating that any point on the perpendicular bisector of a line segment is equidistant from the segment's endpoints

## Central Points in Geometry

### Circumcenter and Incenter

Points formed by the intersection of certain bisectors in a triangle

### Properties and Locations

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

### Additional Points of Concurrency

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

## Practical Applications

### Solving Geometric Problems

The use of the Perpendicular Bisector Theorem and Angle Bisector Theorem to find unknown measurements in triangles

### Importance in Geometry

Understanding these theorems and their converses is crucial in comprehending the properties and structure of triangles