Download Chapter 5.3 Concurrent Lines, Medians, and Altitudes

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:
2
AG  AD
3
2
BG  BE
3
2
CG  CF
3
Theorem 5.9
• The lines that contain the altitudes of a
triangle are concurrent
Classwork/Homework