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Geometry
Notes – Lesson 5.3
Name __________________________________
Concurrent Lines: _______________________________________________________________________________________.
Point of Concurrency: ___________________________________________________________________________________.
Perpendicular Bisectors of the Sides of a Triangle.
Circumcenter of a Triangle:
The Point of Concurrency of the _________________________ __________________________ of a Triangle.
Example. Point _______ is the circumcenter.
Theorem 5-6: The ________________________ ________________________ of the sides of a triangle
are concurrent at a point __________________________ from the ______________________.
Example __________ = ___________ = ___________
By using the circumcenter, you are able to draw a circle
around the triangle going through each vertex.
The circle is ____________________________ about the triangle.
Finding the circumcenter.
Examples:
Find the center of the circle that you can circumscribe about the triangle with vertices
(0, 0)
(-8, 0)
(0, 6)
Find the center of the circle that you can circumscribe about the triangle with vertices
(1, 1)
(5, 1)
(1, 7)
Angle Bisectors of a Triangle.
Incenter of a Triangle:
The Point of Concurrency of the _________________________ __________________________ of a Triangle.
Example. Point _______ is the incenter.
Theorem 5-7: The ____________________ of the angles of a triangle
are concurrent at a point _____________________ from the ___________.
Example __________ = ___________ = ___________
By using the Incenter, you are able to draw a circle
Inside the triangle by measuring from the incenter to a side.
Would you find the circumcenter or the incenter?
a) The towns of Adamsville, Brooksville, and
Cartersville want to build a library
that is equidistant from the three towns.
The circle is ____________________________ in the triangle.
b) City planners want to locate a fountain
equidistant from three straight roads
that enclose a park.
Medians of a Triangle: ___________________________________________________________________________________
__________________________________________________________________________________________.
Centroid of a Triangle:
The Point of Concurrency of the __________________________ of a Triangle.
Example. Point _______ is the centroid.
The centroid is also called the ____________________ _____ ____________________
of a triangle because it is the point where a triangular shape will balance.
Theorem 5-8: The distance from a vertex to the centroid is _______
the distance from each vertex to the midpoint of the opposite side.
DC  23 DJ ,
EC  23 EG ,
FC  23 FH
Also, the distance from the centroid to the opposite side is ________ the
distance from each vertex to the midpoint of the opposite side.
CH  13 FH , CJ  13 DJ , CG  13 EG
Examples.
Altitudes of a Triangle:
________________________________ ________________________ from a vertex to the line containing the opposite side.
***Unlike angle bisectors and medians, an altitude can be a side or it may lie inside or outside the triangle.
Orthocenter of a Triangle:
The point of concurrency of the lines containing the _________________________ of a Triangle.
Theorem 5-9: ___________________________________________________________________________________________
Examples.
a) Is KX a median, an altitude, neither
or both?
b) Is ST a median, an altitude, neither
or both?
c) Is UW a median, an altitude, neither
or both?
Examples.
Is AB an angle bisector, altitude, median or perpendicular bisector?