Download Special Segments in Triangles

Keystone Geometry
» There are four types of segments in a
triangle that create different relationships
among the angles, segments, and vertices.
˃ Medians
˃ Altitudes
˃ Angle Bisectors
˃ Perpendicular Bisectors
Definition of a Median: A segment from the vertex of the
triangle to the midpoint of the opposite side.
Since there are three vertices in every triangle, there are
always three medians.
3
» In acute, right and obtuse triangles the three medians
are drawn inside the triangle.
» To find the median, draw a line from the vertex to the
midpoint of the opposite side.
D
D
D
Special Segments of a triangle:
Altitude
Definition of an Altitude: The perpendicular
segment from a vertex of the triangle to the
segment that contains the opposite side.
5
» To find the altitude, draw a line from the vertex
perpendicular to the opposite side.
» In an acute triangle, the three altitudes are inside
the triangle.
B
D
A
C
» In a right triangle, two of the altitudes are legs of
the triangle and the third altitude is inside the
triangle.
» In an obtuse triangle, two of the altitudes are
outside the triangle and the third altitude is inside
the triangle.
B
A
C
B
B
A
B
A
C
C
A
C
B
B
A
C
A
C
B
A
C
» Draw the three altitudes on the following
triangle:
A
B
C
A
A
B
C
B
C
» We already did this one in Unit 1 Part 1.
» An angle bisector is a line, ray, or segment that divides
an angle into two congruent smaller angles.
ANGLE BISECTOR THEOREM
If a point is on the bisector of an angle, then it is
equidistant from the two sides of the angle.
If AD bisects BAC and DB = AB and DC = AC, then
DB = DC
» What about in a triangle? same thing!
Solve for x.
Because angles
are congruent
and the
segments are
perpendicular,
then the
segments are
congruent.
10 = x + 3
x=7
Solve for x.
Because
segments are
congruent and
perpendicular,
then the angle is
bisected which
means they are
are congruent.
9x – 1 = 6x + 14
3x = 15
x=3
The perpendicular bisector of a segment is a line that is
perpendicular to the segment at its midpoint. The
perpendicular bisector does NOT have to start at a
vertex.
K
In the figure, line l is a perpendicular
bisector of JK.
J
For a perpendicular bisector you must have two things:
Show perpendicularity (90 degree angle)
Show congruence (two equal segments)
 Perpendicular Bisector Theorem
 If a point is on the perpendicular bisector of a
segment, then it is equidistant from the endpoints of
the segment.
 If CP is the perpendicular bisector of AB, then CA =
CB
 Converse of the Perpendicular Bisector Theorem
 If a point is equidistant from the endpoints of the
segment, then it is on the perpendicular bisector of
the segment.
 If DA = DB, then D lies on the perpendicular bisector
of CP.
KG = KH, JG = JH, FG = FH
KG = KH
2x = x + 1
-x -x
x =1
GH = KG + KH
GH = 2x + (x+1)
GH = 2(1) + (1+1)
GH = 2 + 2
GH = 4
Draw the perpendicular bisector of the following
lines, make one a ray, one a line, and one a segment.
X
J
B
K
A
Y
Example:
E
A
A
C
P
M
D
B
B
L
AB
In the scalene ∆CDE, AB is
the perpendicular bisector.
In the right ∆MLN, AB
is the perpendicular
bisector.
Remember, you must show TWO things.
Show perpendicularity and congruence!
N
O
R
In the isosceles
∆POQ, PR is the
perpendicular
bisector.
Q