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:

CFCG

BEBG

ADAG

3

23

23

2

Theorem 5.9

• The lines that contain the altitudes of a triangle are concurrent

Classwork/Homework