Download Unit 6

Unit 5
Bisectors, Medians, and
Altitudes
Part 1
Vocab
• Concurrent Lines
• When three or more lines intersect they
are called concurrent lines
• Point of Concurrency
• The point of intersection for concurrent
lines
Perpendicular Bisector
• A line, segment, or ray that passes through
the midpoint of the side and is perpendicular
to that side.
• Any point on a perpendicular bisector is
equidistant from the endpoints of the
segment.(theorem 5.1)
• Any point that is equidistant from the endpoints of
a segment lies on the perpendicular bisector of
that segment. (theorem 5.2)
• A triangle has three perpendicular bisectors.
Circumcenter
• The intersection of the three
perpendicular bisectors, their point of
concurrency.
• The circumcenter of a triangle is
equidistant from the vertices of a
triangle. (Theorem 5.3)
B
l
k
Lines j, k, and l are
E
j
This means AG = GC = GB
D
C
G
F
A
 bisctors.
Also:
BE = CE & AF = CF & AD = BD
Also: all  bisctors form right
angles with the segments they
bisect.
Angle Bisector Theorems
• Any point on the angle bisector is
equidistant from the sides of the angle.
(theorem 5.4)
• Any Point equidistant from the sides of
an angle lies on the angle bisector.
(theorem 5.5)
Incenter
• The intersection of the three angle
bisectors.
• The incenter of a triangle is equidistant
from each side of the triangle.
C
j
Segments j, k, and l are all
angle bisectors.
3 4
B
F
H
This means BF = DF = EF
G
5
1
Also: <1 = <2, <3 = <4, <5 = <6
k
D
A
6
2
E
I
Also: GH = GI
l
Median
• A segment whose endpoints are a
vertex of a triangle and the midpoint of
the side opposite the vertex.
B
BDis a median of ABC
AD DC
C
A
D
Centroid
• The point of intersection of the three
medians of a triangle
• The centroid is the point of balance for
any triangle
• The centroid is located two thirds of the
distance from a vertex to the midpoint
of the side opposite the vertex on a
median. (Theorem 5.7)
Centroid Example
BD, AF,CEis a median of ABC
therefore:
B
2
1
AG = AF and GH =AF
3
3
F
E
G
2
C
A
D
1
BG = BD and GD = BG
3
3
2
1
CG = CE and GE =CE
3
3
//
//
2
BP  BE
3
2
(10)  BE
3
20
 BE
3
2
3
2(EN )  AE
2(x)  x  17
x  17
1
3
AN  AE  EN
AN  x  17  x
AN  (17)  17  (17)
AN  51
Altitude
• A segment formed from the vertex to the line
containing the opposite side and is
perpendicular to the opposite side.
• Every triangle has three altitudes, their
intersection is know as the orthocenter.
Segment BD is an
altitude, therefore
BDC and BDA are
both right angles
B
C
A
D
Triangle Inequalities
Part 2
Exterior Angle Inequality
Theorem
• If an angle is an exterior angle of a
triangle then its measure is greater than
the measure of either of its corresponding
remote interior angles.(theorem 5.8)
2
m4 > m1
1
3
4
m4 > m2
Triangle Sides and Angles
• If one side of a triangle is longer than
another side, then the angle opposite the
longer side has a greater measure than the
angle opposite the shorter side.
• If one angle of a triangle has a greater
measure than another angle, then the side
opposite the greater angle is longer than the
side opposite the lesser angle.
Example
B
C is the largest angle
5
9
B is the middle angle
C
A is the smallest angle
A
7
E
DFis the largest side
77°
DEis the middle side
65° F
D
38°
EFis the smallest side
Triangle Inequality Theorem
• The sum of the lengths of any two sides
of a triangle is greater than the length
of the third side.
• Always check using the two smallest sides,
they must be larger than the third. If this is
true the numbers will represent a triangle.
Example
• Do these numbers represent a triangle?
1.) 9, 7, 12
Yes
2.) 5, 5, 10
No
3.) 1, 4, 6
No
4.) 6, 6, 2
Yes