Transcript 5.3 Medians and Altitudes of a Triangle B E Geometry Mrs. Spitz Fall 2004 D G A F E Objectives: • Use properties of medians of a triangle • Use properties of altitudes of a triangle 11/6/2015 P.

5.3 Medians and
Altitudes of a
Triangle
B
E
Geometry
Mrs. Spitz
Fall 2004
D
G
A
F
E
Objectives:
• Use properties of medians of a
triangle
• Use properties of altitudes of a
triangle
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Assignment
• pp. 282-283 #1-11, 17-20, 24-26
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Using Medians of a Triangle
In Lesson 5.2, you studied two types of
segments of a triangle: perpendicular
bisectors of the sides and angle
bisectors. In this lesson, you will study
two other types of special types of
segments of a triangle: medians and
altitudes.
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Medians of a triangle
A median of a triangle is a
segments whose
endpoints are a vertex
of the triangle and the
midpoint of the
opposite side. For
instance in ∆ABC,
shown at the right, D is
the midpoint of side
BC. So, AD is a
median of the triangle
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A
MEDIAN
C
D
B
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Centroids of the Triangle
The three medians of a
triangle are concurrent
(they meet). The point
of concurrency is called
the CENTROID OF
THE TRIANGLE. The
centroid, labeled P in
the diagrams in the
next few slides are
ALWAYS inside the
triangle.
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CENTROID
P
acute triangle
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CENTROIDS -
P
centroid
P
centroid
RIGHT TRIANGLE
obtuse triangle
ALWAYS INSIDE THE TRIANGLE
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Medians
• The medians of a triangle have a
special concurrency property as
described in Theorem 5.7. Exercises
#13-16 ask you to use paper folding to
demonstrate the relationships in this
theorem.
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THEOREM 5.7
Concurrency of Medians of a Triangle
The medians of a triangle
intersect at a point that
B
is two thirds of the
distance from each
vertex to the midpoint
of the opposite side.
E
If P is the centroid of
∆ABC, then
AP = 2/3 AD,
BP = 2/3 BF, and
A
CP = 2/3 CE
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C
P
F
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So what?
The centroid of a triangle can be used as
its balancing point. Let’s try it. I’ve
handed out triangle to each and every
one of you. Construct the medians of
the triangles in order to great the
centroid in the middle. Then use your
pencil to balance your triangle. If it
doesn’t balance, you didn’t construct it
correctly.
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Ex. 1: Using the Centroid of a
Triangle
P is the centroid of
∆QRS shown below
and PT = 5. Find R
RT and RP.
S
P
T
Q
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Ex. 1: Using the Centroid of a
Triangle
Because P is the centroid.
RP = 2/3 RT.
R
Then PT= RT – RP = 1/3
RT. Substituting 5 for
PT, 5 = 1/3 RT, so
RT = 15.
Then RP = 2/3 RT =
2/3 (15) = 10
► So, RP = 10, and RT =
15.
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P
T
Q
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Ex. 2: Finding the Centroid of
a Triangle
J (7, 10)
10
Find the coordinates of
the centroid of ∆JKL
N
8
You know that the
centroid is two thirds of
the distance from each
vertex to the midpoint
of the opposite side.
Choose the median KN.
Find the coordinates of
N, the midpoint of JL.
6
L (3, 6)
P
4
M
2
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K (5, 2)
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Ex. 2: Finding the Centroid of
a Triangle
J (7, 10)
10
The coordinates of N are:
3+7 , 6+10 = 10 , 16
2
2
2 2
N
8
Or (5, 8)
6
L (3, 6)
Find the distance from
vertex K to midpoint N.
The distance from K(5,
2) to N (5, 8) is 8-2 or 6
units.
P
4
M
2
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K (5, 2)
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Ex. 2: Finding the Centroid of
a Triangle
J (7, 10)
10
Determine the
coordinates of the
centroid, which is
2/3 ∙ 6 or 4 units up
from vertex K along
median KN.
►The coordinates of
centroid P are (5,
2+4), or (5, 6).
N
8
6
L (3, 6)
P
4
M
2
K (5, 2)
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Distance Formula
I’ve told you before. The distance
formula isn’t going to disappear any
time soon. Exercises 21-23 ask you to
use the Distance Formula to confirm
that the distance from vertex J to the
centroid P in Example 2 is two thirds of
the distance from J to M, the midpoing
of the opposite side.
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Objective 2: Using altitudes of
a triangle
An altitude of a triangle is the
perpendicular segment from the vertex
to the opposite side or to the line that
contains the opposite side. An altitude
can lie inside, on, or outside the
triangle. Every triangle has 3 altitudes.
The lines containing the altitudes are
concurrent and intersect at a point
called the orthocenter of the triangle.
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Ex. 3: Drawing Altitudes and
Orthocenters
•
Where is the orthocenter located in
each type of triangle?
a. Acute triangle
b. Right triangle
c. Obtuse triangle
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Acute Triangle - Orthocenter
B
E
D
G
A
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∆ABC is an acute triangle.
The three altitudes intersect
at G, a point INSIDE the
triangle.
E
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Right Triangle - Orthocenter
∆KLM is a right triangle. The two
legs, LM and KM, are also altitudes.
They intersect at the triangle’s right
angle. This implies that the ortho
center is ON the triangle at M, the
vertex of the right angle of the
triangle.
K
J
M
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Obtuse Triangle - Orthocenter
∆YPR is an obtuse triangle. The three lines that
contain the altitudes intersect at W, a point that is
OUTSIDE the triangle.
P
Z
Y
W
Q
X
R
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Theorem 5.8 Concurrency of Altitudes
of a triangle
The lines containing
the altitudes of a
triangle are
concurrent.
If AE, BF, and CD are
altitudes of ∆ABC,
then the lines AE,
BF, and CD
intersect at some
point H.
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B
H
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E
D
C
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FYI -Exercises 24-26 ask you to use
construction to verify Theorem 5.8. A
proof appears on pg. 838 for your
edification . . .
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