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2.1b: Triangle Properties

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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 ... – PowerPoint PPT presentation

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Title: 2.1b: Triangle Properties


1
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.
GSEs
M(GM)102 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
2
Median
  • Connects a vertex to the of the
    opposite side

MIDDLE
B
Tells us that it cut the side BC in half so BD
DC
F
D
When you combine ALL the medians of one triangle,
you get the CENTROID
A
C
E
A centroid would balance the triangle if you held
it up with a pencil
3
Example
Determine the coordinates of J so that SJ is a
median of the triangle.
Ans Use the midpoint formula for GB
J
J ( 9, 0.5)
J
J
4
Altitude
  • Connects the vertex to where it is perpendicular
    to the opposite side

Combine all three ALTIUDES in a triangle and you
get a ORTHOCENTER
A
Z
T
W
R
P
Tells us the segment is perpendicular
5
Altitude with an obtuse triangle
6
Example
BD is an altitude of Triangle ABC
Find BC, and AC
Ans 7x2090
7x 70
3x-5
X 10
But wait.. Re-read the question
Find BC, so BC 3(10)-5
25
7
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
8
Example
N
Find m NWB if WT is an angle bisector Of
WNB
T
m NWT 3x 8 m NWB 3x 34
3x8
ANS Draw your diagram
W
3x8 3x8 3x 34 6x 16 3x 34 3x
18 x 6 3(6) 34 52
B
9
Perpendicular Bisector
  • A segment that
  • 1) is perpendicular to one side of
    the triangle
  • 2) bisects the same side

H
Draw the segment AT that is a perpendicular
bisector of BO on Triangle HBO
B
O
10
example
  • Name all segments that are (if any)
  • Angle Bisectors
  • Perpendicular Bisectors
  • Altitudes
  • Medians

QU
NONE
RT
SP
11
Example
  • 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

12
Activity
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  • End of notes
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