Transcript 5.1: Special Segments in Triangles

2.1b:
Triangle
Properties
-Special Segments in Triangles
CCSS
G-CO.10
Prove theorems about triangles. Theorems include: measures of
interior angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to the
third side and half the length; the medians of a triangle meet at a point.
GSE’s
M(G&M)–10–2 Makes and defends conjectures, constructs
geometric arguments, uses geometric properties, or uses theorems
to solve problems involving angles, lines, polygons, circles, or right
triangle ratios (sine, cosine, tangent) within mathematics or across
disciplines or contexts
Median
• Connects a vertex to the
opposite side
MIDDLE
of the
B
Tells us that it cut the side BC in half
so BD  DC
F
D
A
E
C
When you combine
ALL the medians of
one triangle, you
get the CENTROID
A centroid would balance the triangle if you held it up with a pencil
Example
Determine the coordinates of J
so that SJ is a median of
the triangle.
Ans: Use the midpoint formula for
GB
J=
J=
J=
 6  12 2   1 


,
2 
 2
 18 1 
 , 
 2 2
9, 0.5
J ( 9, 0.5)
Altitude
Connects the vertex to where it is
perpendicular to the opposite side
A
T
Z
Combine all three ALTIUDE’S
in a triangle and you get a
ORTHOCENTER
W
R
P
Tells us the
segment is
perpendicular
Altitude with an obtuse triangle
Example
BD is an altitude of Triangle ABC
Find BC, and AC
3x-5
Ans: 7x+20=90
7x = 70
X = 10
But wait….. Re-read the question
Find BC, so BC = 3(10)-5
= 25
Angle Bisector
• A segment that cuts one vertex angle in
half and goes to the opposite side of the
triangle
Draw angle bisector AF
B
Indicates the angle A was
bisected
F
A
C
Example
N
Find m  NWB if WT is an angle bisector
Of  WNB
T
m NWT = 3x + 8
3x+8
m NWB = 3x + 34
W
B
ANS: Draw your diagram
3x+8 + 3x+8 = 3x + 34
6x + 16 = 3x + 34
3x = 18
x=6
3(6) + 34 = 52
Perpendicular Bisector
• A segment that:
1) is perpendicular to one side of the
triangle
2) bisects the same side
H
B
Draw the segment AT that is
a perpendicular bisector of BO
on Triangle HBO
O
example
Name all segments that are
(if any)
•Angle Bisectors QU
•Perpendicular Bisectors NONE
•Altitudes RT
•Medians SP
Example
• ABC has vertices A(-4,1) , B ( 1, 6) and
C (3, -4).
Find: 1) The coordinate of T if it is on AB and
TR is a perpendicular bisector to side AB
2) Find the slope of TR
Activity
• End of notes