Chapter 5.3 Concurrent Lines, Medians, and Altitudes

Vocabularies

Vocabularies Concurrent (line) = three or more lines intersect in one point Point of concurrency = the point the concurrent lines intersect Example:

Vocabularies Circumcenter of the triangle = The point of concurrency of the perpendicular bisectors of a triangle Example:

Vocabularies Circumscribed about = A circle is circumscribed about a polygon if the vertices of the polygon are on the circle. A polygon is circumscribed about a circle if all the sides of the polygon are tangent to the circle Examples:

Vocabularies Incenter of the triangle = The incenter of a triangle is the point of concurrency of the angle bisectors of the triangle Example:

Vocabularies Inscribed in = A circle is inscribed in a polygon if the sides of the polygon are tangent to the circle. A polygon is inscribed in a circle if the vertices of the polygon are on the circle Examples:

Vocabularies Median of a triangle = a segment whose endpoints are a vertex and the midpoint of the opposite side Examples:

Vocabularies Centroid = the point of intersection of the medians of that triangle Example:

Vocabularies Altitude of a triangle = the perpendicular segment from a vertex to the line containing the opposite side. Can be inside or outside the triangle. Examples: Acute Triangle Right Triangle Obtuse Triangle

Vocabularies Orthocenter of the triangle = the point of intersection of the lines containing the altitudes of the triangle. Example:

Theorems

Theorem 5.6 The perpendicular bisectors of the sides of a triangle are concurrent at a point equidistant from the vertices Example:

Theorem 5.7 The bisectors of the angles of a triangle are concurrent at a point equidistant from the sides Example:

Theorem 5.8 The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side Example:

Theorem 5.9 The lines that contain the altitudes of a triangle are concurrent

Classwork/Homework