Chapter 5 Notes

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Chapter 5 Notes
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5.1 Perpendiculars and Bisectors
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A perpendicular bisector of a segment is a line
or ray that is perpendicular to the segment at
the midpoint.
D
A
C
B
Perpendicular Bisector Theorem, If a point lies
on the perpendicular bisector of a segment, then
the point is equidistant from the endpoints of
the segment.
Converse of the Perpendicular Bisector Theorem,
If a point is equidistant from the endpoints of a
segment, then the point is on the perpendicular
bisector of the segment.
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Distance from a point to a line is defined to be
the length of the perpendicular segment from the
point to the line (or plane)
Which one represents distance?
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B
C
A
D
Angle Bisector Theorem, If a points lies on the
bisector of an angle, then the point is
equidistant from the sides of the angle.
Converse of the Angle Bisector Theorem, If a
point is equidistant from the sides of an angle,
then the point lies on the angle bisector.
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Constructing a perpendicular to a line through a
given point on the line.
1) From the given point, pick any arc and mark
the circle left and right.
2) Those two marks are your endpoints, and
construct a perpendicular bisector just like the
previously.
Justification. Line is perpendicular by
construction, 3 is on the bisector because it is
equidistant to both endpoints (because radii are
equal), so the line is going through the point.
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Which segment is the perpendicular bisector, how
do you know? Find DK. Find US. Find SK. Find CK
What could DK be so that the segment would NOT be
a perpendicular bisector, how would you know?
U
8
5
D
C
12
S
K
8
R lies on what? How do you know? OM is the
angle bisector of EOT Find MT.
G
E
8
6
M
O
R
T
Y
9
G
a
xo
E
30o
8
6
T
b
yo
O
M
R
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5.2 Bisectors of a Triangle5.3 Medians and
Altitudes of a Triangle
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Where multiple lines meet is called the point of
concurrency. The lines that go through that
point are called concurrent lines.
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The point of concurrency of angle bisectors is
called an INCENTER
Thrm The angle bisectors of triangle intersect
in a point that is equidistant from the three
sides of a triangle.
Justification, points on angle bisector are
equidistant to the sides, then transitive.
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The point of concurrency of perpendicular
bisectors is called a CIRCUMCENTER
Thrm Perpendicular bisectors of the sides of a
triangle intersect in a point that is equidistant
to all the vertices.
Justification, points on perpendicular bisector
are equidistant to the endpoints, then transitive.
So to help keep track of things, its like they
go with the other, angle bisectors equidistant to
sides. Perpendicular bisectors equidistant to
vertices.
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Inside or outside, where do the points of
concurrency meet? Make a sketch and see
CIRCUMCENTERS
INCENTERS
Acute Inside Right On side Obtuse Outside
All inside
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Red lines are angle bisectors.
MA -7x
MB x2 8
A
M
B
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Blue lines are perpendicular bisectors
3x - 4
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5
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Median A line from the midpoint to the vertex
Where they all meet is the CENTROID
The distance from the Centroid to the vertex is
2\3 the median.
The distance from the Centroid to the midpoint is
1\3 the median.
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5
DU
UG
KS
6
9
CU
GS
CS
US
DC
CK
DK
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CM
SM
GM
DS
DG
6
24
GK
CG
DM
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The point of concurrency of altitudes is called
an ORTHOCENTER
Thrm Altitudes all meet at point. Nothing
special about it.
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Inside or outside, where do the points of
concurrency meet? Make a sketch and see
Orthocenters
Centroids
Acute Inside Right On vertex Obtuse Outside
All inside
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5.5 Inequalities in One Triangle
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Terminology and Concepts Terminology ? The side
opposite the angle is the side that is across
from and doesnt touch the angle. Concept ? The
sides and angles opposite from each other often
relate to each other. Angles will use an
uppercase letter, and the side opposite will use
a lower case letter or segment name.
A
c
b
B
C
a
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Theorem 5.10 ? If one side of a triangle is
longer than a second side, then the angle
opposite the first side is larger than the angle
opposite the second side.
R
T
S
Theorem 5.11 ? If one angle of a triangle is
larger than the 2nd angle, then the side opposite
the first angle is longer than the side opposite
the 2nd angle.
Basically, big angle goes with big side, small
angle goes with small side.
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Exterior angle inequality theorem The measure
of an exterior angle of a triangle is greater
than the measure of either of the two nonadjacent
interior angles. NONADJACENT means not attached
to.
R
T
S
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Triangle Inequality Theorem The sum of the
lengths of any two sides of a triangle is greater
than the length of the third side. AB BC gt
AC AC BC gt AB AB AC gt BC
A
B
C
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Pick the greater angle, 1 or 2?
8
11.1
11
9
2
Name the sides, shortest to longest.
R
____ lt ____ lt ____
50o
T
S
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Is it possible for a triangle to have these side
lengths?
Given two side lengths, find the possible lengths
for the 3rd side x
5, 6, 7
5, 6
10, 10, 10
2, 10
1, 1, 2
1, 9
1.1, 1.2, 1.3
4.9, 5, 10
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5.6 Indirect Proof and Inequalities in Two
Triangles
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Hinge Theorem ? If two sides one triangle are
congruent to two sides of another triangle, but
the included angle of the first triangle is
larger than the included angle of the second,
then the third side of the first triangle is
longer than the third side of the second
triangle. Fancy talk for two sides same, one
angle bigger than other, then side is bigger
D
A
E
B
C
F
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Converse of Hinge Theorem ? If two sides one
triangle are congruent to two sides of another
triangle, but the 3rd side of the first triangle
is longer than the 3rd side of the second, then
the included angle of the first triangle is
larger than the included angle of the
second. Fancy talk for two sides same, one sidee
bigger than other, then angle is bigger
D
A
E
B
C
F
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Lots of examples of both types, along with
algebra styles
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List the angles and sides in order
S
S
U
2
U
35o
1
14
45o
30o
D
70o
C
D
70o
13
C
K
K
____ lt ____ ____ lt ____
____ lt ____ lt ____ ____ lt ____ lt ____
student
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Indirect Proof How to write an indirect proof 1.
Assume temporarily that the conclusion is not
true. 2. Reason logically until you reach a
contradiction of the known fact. 3. Point out
the temporary assumption is false, so the
conclusion must be true.
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Practice ? Write the untrue conclusion
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1
a
3
b
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a
1
3
b