Name:_______________________
Notes 5.1-5.4 Triangle Segments
5.1 Midsegment Theorem
Midsegment of a triangle___________________________________________________
________________________________________________________________________
THEOREM 5.1: MIDSEGMENT THEOREM
The segment connecting the midpoints of two sides of a triangle
is _____________ to the third side and is ______ as long as that side.
DE  AC and DE =
1
AC
2
Example 1: Use the Midsegment Theorem to find lengths
Windows A large triangular window is segmented as shown. In the diagram, DF and
EF are midsegments of ABC. Find DF and AB.
Example 2: Use the Midsegment Theorem
In the diagram at the right, QS = SP and PT = TR. Show that QR  ST .
S
5.2 Perpendicular Bisectors
Perpendicular bisector______________________________________________________
_________________________________________________________________________
Equidistant_______________________________________________________________
_________________________________________________________________________
Concurrent_______________________________________________________________________________
Point of concurrency_______________________________________________________________________
THEOREM 5.2: PERPENDICULAR BISECTOR THEOREM
In a plane, if a point is on the perpendicular bisector of a segment, then
it is _____________ from the endpoints of the segment.
If CP is the  bisector of AB , then CA = ______.
THEOREM 5.3: CONVERSE OF THE PERPENDICULAR BISECTOR THEOREM
In a plane, if a point is equidistant from the endpoints of a segment, then
it is on the ___________________________ of the segment.
If DA = DB, then D lies on the _______________________ of AB .
Example 1: Use the Perpendicular Bisector Theorem
AC is the perpendicular bisector of BD . Find AD.
Example 2: Use perpendicular bisectors
In the diagram, KN is the perpendicular bisector of JL .
a. What segment lengths in the diagram are equal?
b. Is M on KN ?
CONCURRENCY OF PERPENDICULAR BISECTORS OF A TRIANGLE


The point of concurrency of the perpendicular bisectors of a triangle
is called the circumcenter.
The circumcenter is equidistant from the three vertices of the triangle.
Example 3: The circumcenter of
ABC is point G. Find GB.
5.3 Angle Bisectors
THEOREM 5.5: ANGLE BISECTOR THEOREM
If a point is on the bisector of an angle, then it is equidistant from the
two _______________ of the angle.
If AD bisects BAC and DB  AB and DC  AC , then DB = _______.
THEOREM 5.6: CONVERSE OF THE ANGLE BISECTOR THEOREM
If a point is in the interior of an angle and is equidistant from the sides of the angle,
then it lies on the _____________ of the angle.
If DB  AB , DC  AC and DB = DC, then AD _________________ BAC.
Example 1: Use the Angle Bisector Theorems
Find the measure of CBE.
Example 2: Use algebra to solve a problem
For what value of x does P lie on the bisector of J ?
CONCURRENCY OF ANGLE BISECTORS OF A TRIANGLE


The point of concurrency of the angle bisectors of a triangle is called the incenter.
The incenter is equidistant from the three sides of the triangle.
1. Find LM.
2. For what value of x does P
lie on the bisector of GFH ?
3. V is the incenter of PQR
Find VS .
5.4 Medians and Altitudes
Median of a triangle________________________________________________________________________
_________________________________________________________________________________________
THEOREM 5.8: CONCURRENCY OF MEDIANS OF A TRIANGLE
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.
The medians of ΔABC meet at P and AP =

2
3
____, BP =
2
3
____, and CP = 2 ____.
3
The point of concurrency of the medians of a triangle is called the centroid of the triangle.
Altitude of a triangle_______________________________________________________________________
_________________________________________________________________________________________
THEOREM 5.9: CONCURRENCY OF ALTITUDES OF A TRIANGLE
The lines containing the altitudes of a triangle are _________________.
The lines containing AF , BE and CD meet at G.



The point of concurrency of the altitudes is called the orthocenter of a triangle.
The altitude is also called the “height” of the triangle.
Sometimes the side of the triangle must be extended and
the altitude is outside of the triangle (obtuse triangle).