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SPECIA
L SEGMENTS
OF
TRIA
NGLES
SE
CT
I ON
S 5
. 2,
5. 3
, 5
. 4
PERPENDICULAR BISECTOR THEOREM
• A point is on the perpendicular bisector if and only if it is equidistant from the
endpoints of the segment.
ANGLE BISECTOR THEOREM
• A point is on the bisector of an angle if and only if it is equidistant from the two sides of the angle.
MEDIANS OF A TRIANGLE
• A median of a triangle is a segment from a vertex to the midpoint of the
opposite side.
ALTITUDES OF A TRIANGLE
• An altitude of a triangle is the perpendicular segment from a vertex to the
opposite side or to the line that contains the opposite side.
CONCURRENCY
The point of intersection of the lines, rays, or segments is called the point of concurrency.
POINTS OF CONCURRENCY
• The point of concurrency of the three perpendicular bisectors a triangle is called the circumcenter.
• The point of concurrency of the three angle bisectors of a triangle is called the incenter.
• The point of concurrency of the three medians of a triangle is called the centroid.
• The point of concurrency of the three altitudes of a triangle is called the orthocenter.
• The incenter and centroid will always be inside the triangle. The circumcenter
and orthocenter can be inside, on, or outside the triangle.
WHAT IS SPECIAL ABOUT THE CIRCUMCENTER?The perpendicular bisectors of a triangle intersect at a point
that is equisdistant
from the vertices of the triangle.
PA = PB = PC
WHAT IS SPECIAL ABOUT THE INCENTER?
• The angle bisectors of a triangle intersect at a point that is equidistant from the
sides of the triangle.
PD = PE = PF
WHAT IS SPECIAL ABOUT THE CENTROID?• The medians of a triangle intersect at a point that is two
thirds of the distance
from each vertex to the midpoint of the opposite side.
WHAT IS SPECIAL ABOUT THE ORTHOCENTER?• There is nothing special about the point of concurrency of
the altitudes of a
triangle.
ASSIGNMENT
Pg. 306 #3, 5, 11-17 odds
Pg. 313 #3-23 odds
Pg. 322 #3-7odds, 17-21 odds, 33, 35