Download Concurrent Lines, part 1

Daily Warm-Up Quiz
1. Which of your classmates disclosed to
a teacher that Mrs. M. sometimes
refers to Makenna as Mackenzie…and
vice versa?
2. Who told this same teacher that
period 2 Geometry is my “favorite
class”? How did you determine this?
3. Who shared that since Monday, Kaylin
has been renamed “Kylin”?
Mrs. McConaughy
Geometry
1
Relationships in Triangles
Concurrent Lines, Medians and
Altitudes
Mrs. McConaughy
Geometry
2
Part I: Identifying
Properties of Angle
Bisectors and
Perpendicular Bisectors in
Triangles
Mrs. McConaughy
Geometry
3
In this lesson, we will
identify
properties of
perpendicular bisectors
and angle bisectors in
triangles.
∆ OPS
Mrs. McConaughy
Geometry
4
Long before the first pencil and paper, some
curious person drew a triangle in the sand and
bisected the three angles. He noted that the
bisectors met in a single point and decided to
repeat the experiment on an extremely
obtuse triangle. Again, the bisectors
concurred. Astonished, the person drew yet a
third triangle, and the same thing happened
yet again! Unlike squares and circles, triangles
have many centers. The ancient Greeks found
four: incenter, centroid, circumcenter, and
orthocenter.
Triangle Centers: http://faculty.evansville.edu/ck6/tcenters/index.html
Mrs. McConaughy
Geometry
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Vocabulary & Key Concepts
When three or more lines intersect in one
concurrent
point, they are called _____________.
The point at which they intersect is called
point of concurrency
the _________________.
Mrs. McConaughy
Geometry
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Vocabulary and Key Concepts
The point of
concurrency of the
angle bisectors of a
triangle is called the
incenter of the
_________
triangle.
I is the incenter
of the ∆.
THEOREM: The bisectors of the angles of
a ∆ are concurrent at a point (incenter)
equidistant from the sides.
Mrs. McConaughy
Geometry
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Checking for Understanding
City planners want
to locate a
fountain
equidistant from
three straight
roads that
enclose a park.
Explain how they
can find the
location.
Mrs. McConaughy
Geometry
Andover Road
Check your solution here!
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Alert! The common distance is the radius of a circle that passes
through the vertices.
Vocabulary and Key Concepts
The point of
concurrency of the
perpendicular
bisectors of a
triangle is called the
____________
circumcenter
of
the triangle.
O is the circumcenter.
THEOREM: The perpendicular bisectors of
the angles of a ∆ are concurrent at a point
(circumcenter) equidistant from the
vertices.
Mrs. McConaughy
Geometry
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Checking for Understanding: Finding
Checking
for Understanding
the
Circumcenter
Find the center of
the circle that you
can circumscribe
about ∆ OPS.
Solution: Two
perpendicular
bisectors of the
sides of ∆ OPS are x =
2 and y = 3. These lines
intersect at (2,3). This
point is the center of
Mrs. McConaughy
Geometry
the circle.
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Homework
Mrs. McConaughy
Geometry
11
Part II: Identifying Properties
of Medians and Altitudes in
Triangles
Mrs. McConaughy
Geometry
12
In this lesson, we will
identify
properties of medians
and altitudes in triangles.
∆ OPS
Mrs. McConaughy
Geometry
13
Median of a Triangle
A median of a
triangle is a
segment whose
endpoints are a
vertex and the
midpoint of the
opposite side.
Vertex
Midpoint
Mrs. McConaughy
Geometry
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Vocabulary and Key Concepts
The point of
concurrency of the
medians of a triangle
is called
centroid
the___________
of the triangle.
G is the centroid.
FG = 2/3 FC EG = 2/3 EB AG = 2/3 AD
Theorem: The medians of a triangle are concurrent at
a point that is two-thirds the distance from each
vertex to the midpoint of the opposite side.
Mrs. McConaughy
Geometry
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Checking for Understanding
Finding the Lengths of Medians.
G is the centroid.
G is the centroid of
∆ ABC and DG = 6.
Find AG.
AG = 2/3 AD;
DG = 1/3 AD
6 = 1/3 AD
18 = AD
Mrs. McConaughy
Geometry
16
Altitude of a Triangle:
An altitude of a triangle is the perpendicular
segment from a vertex to the line
containing the opposite side.
Unlike angle bisectors and medians, an
altitude can lie inside, on, or outside the
triangle.
Acute Triangle:
Mrs. McConaughy
Interior
Altitude
Right Triangle: Obtuse Triangle:
Geometry
Altitude
is a side Exterior Altitude17
Altitude of a Triangle
The lines containing
the altitudes of a
triangle are
concurrent at the
orthocenter.
Theorem: The lines that contain the
altitudes of a triangle are concurrent.
Mrs. McConaughy
Geometry
http://www.mathopenref.com/triangleorthocenter.html
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Identifying Medians and
Altitudes
A
Is CM a
median,
altitude, or
neither?
Explain.
M
B
H
Is BH a median, altitude,
or neither? Explain.
Mrs. McConaughy
Geometry
C
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Homework
Mrs. McConaughy
Geometry
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Solution: City Planning Dilemma
The roads form a triangle around the
park. By our new theorem, we know
bisectors of the angles of a
that the __________________
triangle are concurrent at a point
_________
equidistant from the sides. The city
planners should find the point of
of the angles
concurrency of the bisectors
_______________
of the triangle formed and locate the
fountain there.
Mrs. McConaughy
Click here to return to the lesson!
Geometry
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