Transcript Chapter 5 Notes - Troy High School

Chapter 5 Notes
5.1 – Perpendiculars and
Bisectors
A perpendicular bisector of a segment is a line or ray that is
perpendicular to the segment at the midpoint.
D
A
If D is on  bisector
then AD  CD
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.
C
B
If AD  CD
then D is on  bisector
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.
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?
B
A
C
D
If C is on  bisector
If BC  DC
then BC  DC
then C is on  bisector
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.
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.
Which segment is the
perpendicular bisector, how
do you know?
What could DK be so that
the segment would NOT be
a perpendicular bisector,
how would you know?
Find DK.
Find US.
Find SK.
U
8
5
D
C
S
Find CK
K
R lies on what? How do you
know?
OM is the angle bisector of
EOT
Find MT.
G
8
E
6
M
O
R
T
Y
G
E
8
O
a
xo
30o
6
yo
M
T
b
R
5.2 – Bisectors of a Triangle
5.3 – Medians and Altitudes of a Triangle
Where multiple lines meet is called the point of
concurrency. The lines that go through that point are
called concurrent lines.
The point of concurrency
Thrm: The angle bisectors of
triangle intersect in a point that of angle bisectors is called
an INCENTER
is equidistant from the three
sides of a triangle.
Justification, points on angle
bisector are equidistant to the
sides, then transitive.
Thrm: Perpendicular bisectors
of the sides of a triangle
intersect in a point that is
equidistant to all the vertices.
The point of concurrency
of perpendicular bisectors
is called a CIRCUMCENTER
Justification, points on
perpendicular bisector are
equidistant to the endpoints,
then transitive.
So to help keep track of things, it’s like they go
with the other, angle bisectors equidistant to
sides. Perpendicular bisectors equidistant to
vertices.
Inside or outside, where do the points of concurrency meet?
Make a sketch and see
CIRCUMCENTERS
Acute – Inside
Right – On side
Obtuse – Outside
INCENTERS
All inside
Red lines are angle bisectors.
MA = -7x
MB = x2 – 8
M
B
A
Blue lines are perpendicular bisectors
5
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.
D
Think
2 :1
M
U
G
C
DU = 5
CU =
DC =
SM =
DS = 24
DM =
K
KS = 9
CS =
CK =
CM = 18
GM =
CG =
S
UG =
GS =
US =
DK =
6
DG =
GK =
6
Thrm: Altitudes all meet at
point.
Nothing special about it.
The point of concurrency
of altitudes is called an
ORTHOCENTER
Inside or outside, where do the points of concurrency meet?
Make a sketch and see
Orthocenters
Acute – Inside
Right – On vertex
Obtuse – Outside
Centroids
All inside
5.5 – Inequalities in One Triangle
Terminology and Concepts
Terminology  The side opposite the angle is the side that is across
from and doesn’t 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
a
C
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
Given : RST ; RT  RS
Prove: mS  mT
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.
Given : RST ; mS  mT
Prove: RT  RS
Basically, big angle goes with big side, small angle goes with small side.
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
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 > AC
AC + BC > AB
AB + AC > BC
A
B
C
Pick the greater angle, 1 or 2?
8
11.1
11
9
2
Name the sides, shortest to longest.
R
____ < ____ < ____
50o
T
S
Is it possible for a triangle to have Given two side lengths, find the
these side lengths?
possible lengths for the 3rd side ‘x’
5, 6, 7
10, 10, 10
1, 1, 2
1.1, 1.2, 1.3
4.9, 5, 10
5, 6
2, 10
1, 9
5.6 – Indirect Proof and
Inequalities in Two Triangles
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
B
Given : BA  ED, BC  EF,
mB  mE
T hen: AC  DF
E
C
F
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
B
Given : BA  ED, BC  EF,
AC  DF
T hen: mB  mE
E
C
F
Lots of examples of both types, along with algebra
styles
List the angles and sides in order
S
S
U
30o
D
U
35o
2
1
14
45o
D
C
K
13
70o
70o C
K
____ < ____ < ____
____ < ____
____ < ____ < ____
____ < ____
student
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.
Practice  Write the untrue conclusion
Prove : n is odd
Prove: sum of interior
angles of a triangleis 180
Prove: AB  AB
Prove: Mr. Kim is a genius
a
b
1
3
Given : m1  m3
Prove: a || b
Given : m1  50, m2  60
Prove: 1 and 2 are not a linear pair
a
b
Given : 1, 3 not supp.
1
3
Prove: a || b