5.2 Flashcards
Perpendicular Bisector of a Triangle
Is a segment that is part of a perpendicular bisector of one of the sides.
Angle Bisector of a Triangle
Is a segment that bisects one of the angles of the triangle.
Median
Is a segment whose endpoints are a vertex and the midpoint of the opposite side.
Altitude
Is a segment from a vertex that is perpendicular to the opposite side or to the line containing the opposite side. ***An altitude may lie inside or outside the triangle.
Concurrent Lines
A set of lines that share the same point of intersection.
Circumcenter
Is equidistant from the three vertices of the triangle (and is the center of the circumscribed circle).
Incenter
Is equidistant from the three sides of the triangle (and is the center of the inscribed circle).
Centroid
Is two-thirds of the distance from the vertex to the midpoint of the opposite side.
Orthocenter
The lines containing the altitudes of a triangle are concurrent in the orthocenter of a triangle.
Point of intersection of 3 altitudes in a triangle.
Thm. 5.5: Concurrency Properties
The 3 perpendicular bisectors, the 3 angle bisectors, the 3 medians, and the 3 altitudes are all concurrent.
or. ..
a. The lines containing the perpendicular bisectors of a triangle are concurrent. Their common point is the circumcenter of the triangle. The circumcenter is equidistant from the vertices of the triangle.
b. The angle bisectors of a triangle are concurrent. Their common point is the incenter of the triangle. The incenter is equidistant from the three sides of a triangle.
c. The medians of a triangle are concurrent. Their common point is the centroid of the triangle. The centroid is two-thirds of the distance from each vertex to the midpoint of the opposite side.
d. The lines containing the altitudes of a triangle are concurrent. Their common point is the orthocenter of a triangle.
