Triangle_Bisectors,_Medians_and_Altitutdes_5.1,_5.2_(web)

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CHAPTER 5.1
Check.4.14 Identify and use medians, mid-segments,
altitudes, angle bisectors, and perpendicular bisectors of
triangles to solve problems (e.g., find segment lengths,
angle measures, points of concurrency).
POINTS OF CONCURRENCY

The point where the Perpendicular bisectors of a
triangle meet is called the _______________.


The point where the Angle Bisectors of a triangle
meet is called the _______________.


The ___________of a triangle is equidistant from the
__________of the triangle.
The point where the Medians of a triangle meet is
called the _______________.


The __________ of a triangle is ___________from the of the
_________triangle.
The ________of a triangle is located _________of the
distance from a _________ to the _________ of the side
opposite the vertex.
The point where the Altitudes of a triangle meet is
called the _______________.
PERPENDICULAR BISECTORS - THEOREMS
Any point on the
perpendicular bisector
of a segment is
equidistant from the
endpoints of the
segment.

ABCD and AB
bisects CD then AC =
AD and BC = BD

A
D
C
B
PERPENDICULAR BISECTORS - THEOREMS
Converse
 Any point equidistant from
the endpoints of a segment
lies on the perpendicular
bisector of the segment



If AC = AD, then A lies on
the perpendicular bisector
of CD.
If BC = BD then B lies on
the perpendicular bisector
of CD.
A
D
C
B
USE THE PERPENDICULAR BISECTOR
THEOREMS
A. Find BC.
BC = AC
Perpendicular Bisector Theorem
BC = 8.5
Substitution
Answer: 8.5
USE THE PERPENDICULAR BISECTOR
THEOREMS
B. Find XY.
Answer: 6
CIRCUMCENTER THEOREM

The point where the Perpendicular bisectors of a
triangle meet.
B
AJ = CJ
 AJ = BJ
 CJ = BJ
 Transitive property


J
A
Circumcenter of a triangle is equidistant from
the vertices of the triangle.
C
POINTS OF CONCURRENCY

The point where the Perpendicular bisectors of a
Circumcenter
triangle meet is called the _______________.


The point where the Angle Bisectors of a triangle
Incenter
meet is called the _______________.


The ___________of a triangle is equidistant from the
__________of the triangle.
The point where the Medians of a triangle meet is
Centroid
called the _______________.


Circumcenter
equidistant
The __________
of a triangle is ___________from
the of the
_________triangle.
verticies
The ________of a triangle is located _________of the
distance from a _________ to the _________ of the side
opposite the vertex.
The point where the Altitudes of a triangle meet is
called the _______________.
Orthocenter
ANGLE BISECTORS

Any point on the angle bisector is equidistant
from the sides of the angle.
B
J


Converse
A
Any point equidistant from the sides of an angle
lies on the angle bisector
C
USE THE ANGLE BISECTOR THEOREMS
A. Find DB.
DB = DC
Angle Bisector Theorem
DB = 5
Substitution
Answer: DB = 5
USE THE ANGLE BISECTOR THEOREMS
B. Find mWYZ.
USE THE ANGLE BISECTOR THEOREMS
WYZ  XYW
Definition of angle bisector
mWYZ = mXYW
Definition of congruent angles
mWYZ = 28
Substitution
Answer: mWYZ = 28
INCENTER THEOREM

The point where the Angle Bisectors of a triangle
meet.
B
J
A

The incenter of a triangle is equidistant from the
sides of the triangle.
C
POINTS OF CONCURRENCY

The point where the Perpendicular bisectors of a
triangle meet is called the _______________.
Circumcenter


The point where the Angle Bisectors of a triangle
Incenter
meet is called the _______________.


The ___________of
incenter
a triangle is equidistant from the
sides
__________of
the triangle.
The point where the Medians of a triangle meet is
Centroid
called the _______________.


TheCircumcenter
__________ of a triangle is equidistant
___________from the of the
_________triangle.
verticies
The ________of a triangle is located _________of the
distance from a _________ to the _________ of the side
opposite the vertex.
The point where the Altitudes of a triangle meet is
called the _______________.
Orthocenter
USE THE INCENTER THEOREM
A. Find ST if S is the
incenter of ΔMNP.
By the Incenter Theorem, since S is
equidistant from the sides of ΔMNP,
ST = SU.
Find ST by using the
Pythagorean Theorem.
a2 + b2 = c2
Pythagorean Theorem
82 + SU2 = 102
Substitution
64 + SU2 = 100
82 = 64, 102 = 100
SU2 = 36
Subtract 64 from each side.
SU = ±6
Take the square root
of each side.
Since length cannot be negative, use only the positive square root, 6. Since
ST = SU, ST = 6.
Answer: ST = 6
The Centroid is the point of balance for any triangle.
CENTROID THEOREM
The point where the Medians of a triangle meet.
B
 Median connects the

Midpoint of the side with
 the vertex of the opposite angle

D
A

E
L
F
The centroid of a triangle is located two thirds of
the distance from a vertex to the midpoint of the
side opposite the vertex.

AL= 2/3 AE, BL=2/3BF and DL=2/3DC
C
POINTS OF CONCURRENCY

The point where the Perpendicular bisectors of a
triangle meet is called the Circumcenter
_______________.


The point where the Angle Bisectors of a triangle
Incenter
meet is called the _______________.


The ___________of
incenter
a triangle is equidistant from the
sides
__________of
the triangle.
The point where the Medians of a triangle meet is
Centroid
called the _______________.


equidistant
TheCircumcenter
__________ of a triangle is ___________from
the of the
_________triangle.
verticies
2/3
Centroid
The ________of
a triangle is located ____of
the distance
vertex
midpoint
from a __________
to the ___________
of the side opposite
the vertex.
The point where the Altitudes of a triangle meet is
called the _______________.
Orthocenter
USE THE CENTROID THEOREM
In ΔXYZ, P is the centroid and YV = 12. Find YP
and PV.
Centroid Theorem
YV = 12
Simplify.
YP + PV
= YV
8 + PV
= 12
PV
=4
Segment Addition
Subtract 8 from each side
Answer: YP = 8; PV = 4
Points S, T, and U are the
midpoint of DE, EF and DF
respectively.
 Find x, y, and z
 Centroid Theorem

CALCULATIONS
E
S
6
D
y

T
4z A 2x-5
2.9
U

4.6


F




Centroid is located at 2/3 the
length of the median
EA=2/3EU
DT=DA+AT
Y=2/3(y+2.9)
DT=2x-5+6=2x+1 3Y = 2y + 5.8
1y=5.8
DA=2/3DT
6=2/3(2x+1)
FA=2/3FS
3/2(6)=2x+1
4.6=2/3(4z+4.6)
9=2x+1
13.8=8z + 9.2
X=4
4.6 = 8z
Z=0.575
BASEBALL A fan of a local baseball team is
designing a triangular sign for the upcoming
game. In his design on the coordinate plane,
the vertices are located at (–3, 2), (–1, –2), and (–
1, 6). What are the coordinates of the point
where the fan should place the pole under the
triangle so that it will balance?
A.
B.
C. (–1, 2)
D. (0, 4)
FIND THE ORTHOCENTER OF JKL
Find Altitude from J to KL
 LK
 Slope LK is -1/3, m=
3
 Point slope (1,3) m= 3
 y-3=3(x-1)
 y-3=3x-3
 y = 3x
 Find Altitude from K to JL
 Slope of JL is 3/2, m=-2/3
 y+1=-2/3(x-2)
 y+1=-2/3x+4/3
 y = -2/3x+1/3
J(1,3)

L(-1,0)








K(2,-1)
y = 3x
y = -2/3x+1/3 find intersection
3x= -2/3x+1/3 multiply by 3
9x =-2x + 1
11x = 1
x = 1/11
y=3(1/11)
y=3/11
PRACTICE ASSIGNMENT

Page 327, 10 – 30 4th,

Page 338, 6 – 30 4th

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