Special line segments in triangles
The first page gives a quick overview about what each of these are and
properties of these special line segments. The second and third page give
detailed instructions on how to construct them. It is hard to give clear directions
on how to do this without using a picture as a reference. Look in your book on
page 522 and 523 for detailed pictures – if you have any questions or notice any
mistakes, please email me.
• Median
-- A line segment that connects a vertex to the midpoint of the opposite side
-- There are 3 medians in a triangle. These 3 line segments are concurrent.
-- The point where the 3 medians meet is called the centroid of the circle.
This is the center of gravity of the triangle (balancing point of triangle)
• Angle bisector
-- A line segment that bisects an angle of a triangle (cuts angle in half)
-- There are 3 angles bisectors in a triangle. These 3 line segments are
concurrent.
-- The point where the 3 angle bisectors meet is called the incenter of the
triangle. This point is equidistant from each of the 3 sides of the triangle.
-- You can draw a circle so that the circle touches each side of the sides of
the triangle and the incenter is the center of the circle, and the circle will be
inside the triangle. This circle is inscribed in the triangle.
• Perpendicular bisector
-- A line that goes through the midpoint of a side and is perpendicular to that
side.
-- There are 3 perpendicular bisectors in a triangle. These 3 lines are
concurrent.
-- The point where the 3 perpendicular bisectors meet is called the
circumcenter of the triangle. This point is equidistant from each of the 3
vertices of triangle.
-- You can draw a circle where each of the 3 vertices is on the circle and the
circumcenter is the center of the triangle, and the triangle will be inside the
circle. This circle circumscribes the triangle.
• Altitude
-- A perpendicular line segment that connects a vertex to the side opposite
that vertex.
-- There are 3 altitudes in a triangle. These 3 line segments are concurrent.
-- The point where the 3 altitudes meet is called the orthocenter of the
triangle.
Note: These 4 points are only going to be the same in the case of an equilateral triangle.
Although, no matter what type of triangle you have, 3 of the points (centroid,
circumcenter, and orthocenter) will always be collinear. The line drawn through these 3
points is called the Euler line, after Leonard Euler.
• Median: a line segment drawn from a vertex to the side opposite the vertex; the
line segment intersects the side of the triangle at its midpoint – the point that
divides the side in half
-- To construct a median using a protractor, you must follow these steps:
• Measure the side of the triangle to which you are drawing a line segment
• Based on the measurement, divide the side of the triangle into 2 equal
pieces (in half)
• Now draw a line segment from the vertex opposite the side you
measured to the half-way point of the side
• Repeat this process for all 3 sides of the triangle.
• Note: Make sure to label your median correctly. It is a line segment and
therefore you must use correct notation. For example, you would say
_
something like AC is a median of triangle BAF .
• Angle bisector: a line segment drawn from a vertex that bisects (cuts the angle
into 2 equal pieces) the angle formed by that vertex.
-- To construct an angle bisector using a protractor, you must follow these steps:
• Measure the angle that you want to bisect (cut in half) with a protractor;
you will have to extend the lines of the triangle in order to get an accurate
measurement of the angle.
• Based on the measurement, divide the angle into 2 equal pieces.
• Now draw a line segment from the vertex to the point that you marked to
denote the 2 equal pieces of the angle.
• Repeat this process for all 3 angles of the triangle.
• Note: Make sure to label your angle bisector correctly. It is a line
segment and therefore you must use correct notation. For example, you
_
would say something like AC is an angle bisector of triangle BAF .
• Perpendicular bisector: a line drawn that goes through the midpoint of a side
of the triangle; the line drawn must be perpendicular to the side of the triangle
(i.e., the line forms a 90° angle at the midpoint).
-- To construct a perpendicular bisector using a protractor, you must follow these
steps:
• Measure the side of the triangle through which you want to draw the
perpendicular line.
• Based on the measurement, divide the side of the triangle into 2 equal
pieces.
• From the point on the side of the triangle that denotes the midpoint of
that side, form a right angle (use a protractor!) with the side of the triangle
being one side of the right angle. Make a mark where the other side of the
right angle will be.
• Using the 2 points you now have marked (the midpoint and where the
other side of the right angle will be), draw a line. This line is your
perpendicular bisector.
• Repeat this process for all 3 sides of the triangle.
• Note: Make sure to label your perpendicular bisector correctly. It is a line
and therefore you must use correct notation. For example, you would say

_
_
something like AC is an perpendicular bisector of side PN (where PN is
a side of the triangle).
• Altitude: a line segment that is drawn from a vertex to the side of the triangle
opposite the vertex. This line segment is perpendicular (form a 90° angle) with
the side of the triangle opposite the vertex. NOTE: Some altitudes may be
drawn outside of the triangle, whereas some altitudes are just the sides of the
triangle itself.
-- To construct an altitude using a protractor, you must follow these steps:
• From the vertex that you want to draw an altitude from, draw a line
that is tangent to that vertex (line touches the triangle ONLY at the vertex)
and parallel to the side of the triangle opposite the vertex.
• Take a protractor and use that vertex as the vertex of your right angle
and the tangent line as one of the sides of your right angle. Make a mark
where the other side of your right angle will be.
• Using the 2 points you now have (the vertex and the one you made for
where the other side of the right angle will be),draw a line. This line is
your is your altitude.
• Repeat this process for all 3 vertices of the triangle.
• Note: Make sure to label your altitude correctly. It is a line segment and
therefore you must use correct notation. For example, you would say something
_
like AC is an altitude of triangle BAF .