4.7 Medians, Altitudes, and Perpendicular Bisectors

Median of a Triangle

A segment that connects the vertex of a triangle to the midpoint of the opposite side

Centroid

The point of concurrency of the medians of a triangle, cuts medians into 2:1 ratio, always inside triangle

Altitude of a Triangle

A segment that has one endpoint at the vertex of a triangle and is perpendicular to the line containing the opposite side

Orthocenter

The point of concurrency of the altitudes of a triangle, no special properties, inside triangle for acute, on triangle for right, and outside triangle for obtuse

Perpendicular Bisectors of a Triangle

A segment that passes through the midpoint of a side of a triangle and is perpendicular to that side (DON'T USE VERTEX)

Circumcenter

the point of concurrency of the perpendicular bisectors of a triangle, equidistant from vertices, inside triangle on acute, on triangle on right, and outside triangle on obtuse

Angle Bisector of a Triangle

A segment that has one endpoint at a vertex of a triangle and divides an angle into two congruent angles

Incenter

the point of concurrency of the angle bisectors of a triangle, equidistant from sides, always inside triangle

A point of concurrency is…

a point where lines of concurrency meet

If a point lies on the perpendicular bisector of a segment…

then the point is equidistant from the endpoints of the segment

If a point is equidistant from the endpoints…

then the point lies on the perpendicular bisector of the segment

If a point lies on the bisector of an angle

then the point is equidistant from the sides of the angle

If a point is equidistant from the sides of an angle,

then the point lies on the bisector of each angle