Notes – Lesson 5.3
Geometry
Name __________________________________
When three or more lines intersect in one point, they are ______________________________.
The point at which they intersect is the ________________ _____ _________________________________.
Perpendicular Bisectors of the Sides of a Triangle.
1. Each side of the triangle has been bisected into 2 equal parts.
2. The point of concurrency of the perpendicular
bisectors of a triangle is called the
_____________________________
Example. Point _______ is the circumcenter.
3. Theorem: The perpendicular bisectors of the sides of a triangle are concurrent at a point __________________________
from the______________________.
Example. (from the 2nd picture above)
__________ = ___________ = ___________
4. By using the circumcenter, you are able to draw a circle around the triangle going through each vertex. The circle is
______________________________ about the triangle.
5. Finding the circumcenter.
Example:
Find the center of the circle that you can circumscribe about the triangle with vertices (0, 0), (-8, 0), and (0, 6).
Find the center of the circle that you can circumscribe about the triangle with vertices (1, 1), (5, 1), and (1, 7).
Angle Bisectors of a Triangle.
1. Each angle has been bisected into two equal angles.
2. The point of concurrency of the angle
bisectors of a triangle is called the
_____________________________
Example. Point _______ is the incenter.
3. Theorem. The bisectors of the angles of a triangle are concurrent at a point _____________________ from the ___________.
Example. (from the 2nd picture above)
__________ = ___________ = ___________
4. By using the incenter, you are able to draw a circle inside the triangle by measuring from the incenter to a side. The circle is
____________________ in the triangle.
5. 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.
b) City planners want to locate a fountain
equidistant from three straight roads
that enclose a park.
Medians of a Triangle.
1. The median of a triangle is a segment whose endpoints
are a vertex and the midpoint of the opposite sides.
2. The point of concurrency of the medians of a
triangle is called the
_____________________________
Example. Point _______ is the centroid.
3. The centroid is also called the ____________________ _____ ____________________ of a triangle because it is the point
where a triangular shape will balance.
4. Theorem
The distance from a vertex to the centroid is _______ the distance
from each vertex to the midpoint of the opposite side.
EC  23 EG ,
DC  23 DJ ,
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
5. Examples.
M is the centroid of  WOR.
a) If WM = 16, find WX
b) If MY = 6, find OY
c) If MR = 30, find ZR
Altitudes of Triangles
1. An altitude of a triangle is the ________________________________ ________________________ 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 outside the triangle.
2. 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?
3. Examples.
Is AB an angle bisector, altitude, median or perpendicular bisector?