Welcome to Chapter 5 Here we will talk about new relationships in triangles' Get ready for some voca

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Welcome to Chapter 5! Here we will talk about
new relationships in triangles. Get ready for
some vocabulary! The plan for the week is
this Monday 5-1 Bisectors, Medians, and
Altitudes (part I) Tuesday Vocab. Quiz 5-1
(part II) Wednesday 5-2 Inequalities and
Triangles Thursday 5-4 The Triangle
Inequality Friday 5-5 Inequalities Involving
Two Triangles
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OK! To begin the chapter, we will talk about
bisectors first. There are two types of
bisectors in triangles
Angle Bisectors
Perpendicular Bisectors
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Lets start with angle bisectors. Watch as the
triangle below has bisectors drawn through its
angles. What do you notice
about the three angle bisectors?
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The angle bisectors intersect at one common
point. When three or more lines intersect at a
common point, the lines are called concurrent
lines, and their point of intersection is called
the point of concurrency. Concurrent
lines three or more lines that intersect at
a common point Point of concurrency point
where concurrent lines intersect Lets keep
going with our diagram. What else can we say?
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The blue lines are drawn from the intersection
point of the three angle bisectors so that they
are perpendicular to the sides of the triangle.
Once we draw the blue lines, we see that those
points (point E, for example) fall exactly on the
green circle, which fits perfectly in the
triangle. What name should we give point D?
INCENTER
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Incenter

• The angle bisectors of a triangle are concurrent,
  and their point of concurrency has a special
  name, called the incenter.
• Incenter point of concurrency for angle
  bisectors
• The incenter is equidistant from the sides of the
  triangle.

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Each triangle has three angle bisectors. But a
triangle also has three perpendicular bisectors.

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The blue lines are drawn are three perpendicular
bisectors so that they are perpendicular to the
sides of the triangle and bisect the sides.
These are different from the perpendicular lines
we drew with angle bisectors because these
perpendicular lines divide the sides into
congruent parts. These, too, are concurrent
linesand meet at a point of concurrency. What
name should we give point O?
Circumcenter
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As with angle bisectors, we can draw a green
circle again. But this time there will be
something different. What is
different about the green circle this time? Last
time the incenter was equidistant from points on
the triangle. What points is O equidistant from
now?
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Instead of being inside the triangle, this time
the circle is outside. Circumcenter equidistant
from all of the vertices of the triangle.
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A couple more theorems dealing with
bisectors Theorem 5.4 Any point on the angle
bisector is equidistant from the sides of the
angle Now BACKWARDS Theorem 5.5 Any point
equidistant from the sides of an angle lies on
the angle bisector
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Medians are also involved in triangles. A median
is a segment whose endpoints are a vertex of a
triangle and the midpoint of the side opposite
the vertex. The medians in a triangle also
intersect at a common point. The point of
concurrency for medians is called a
centroid. AE, BF, and DC are the medians
of triangle ABC.Point L is the centroid.
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Theorem 5.7 In a triangle, the distance from a
vertex to a centroid is two thirds the distance
from the vertex to the midpoint of the opposite
side. Example If L is the centroid
of triangle ABC, AL (2/3)AE, BL (2/3)BF,
and CL (2/3)CD.
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Lets look at an example together. Points S, T,
and U are the midpoints of DE, EF, and DF,
respectively. Find x, y, and z. DT DA
AT 6 (2x 5) 2x
1 DA (2/3) DT 6 (2/3) (2x
1) 18 4x 2 16 4x 4
x Then do the same for y and z.
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What have we talked about today? Incenters
and circumcenters are related to the sides and
angles of triangles. The incenter is the
intersection of angle bisectors and is
equidistant from the sides of the triangle.
The circumcenter is the intersection of
perpendicular bisectors and is equidistant from
the vertices. The centroid is the intersection
of medians. The distance from vertex to centroid
is two thirds that of the distance from vertex to
opposite sides midpoint.
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Your homework tonight deals with these concepts.
Homework Page 243, 16, 17, 19, 20, 21, 24
29 all AND DONT FORGET Your vocabulary quiz
is tomorrow!