6-1 Medians

Report
6-1 Medians
Objective: To identify and construct
medians in triangles
February 2011
Fellow: Brooke Odle
Teacher: Ms. Sanchez
Saint Vincent Academy
Lesson Overview
• Introduce topic
– Vocabulary and Theorem 6-1
• Example Problems
• Real World Application
– Electrocardiograms
• Class work
– Clicker Questions
What is a median?
B
median BD
• In a triangle, a median
is a segment that joins
a vertex of the triangle
and the midpoint of
the side opposite that
vertex.
• A triangle has 3
medians.
median TU
C
R
median ZW
Example 1
A
• In
ABC, CE and AD
are medians.
• Find BE if AB = 18.
• ANSWER: 9
Example 2
A
• In
ABC, CE and AD
are medians.
• If CD = 2x+5, BD = 4x-1,
and AE = 5x-2, find BE.
• ANSWER: 13
Intersections of Medians
• The medians of triangle
K
JKM, JR, KP, and MQ,
Q
intersect at a common
point called the
J
centroid.
R
X
• When 3 or more lines or
P
segments meet at the
M
same point, the lines
X is the centroid of
JKM
are concurrent.
and JR, KP, and MQ are all
concurrent.
Relationship between Lengths
• There is a relationship
between the length of
the segment from the
vertex to the centroid
and the length of the
segment from the
centroid to the
midpoint.
C
E
A
13
26
D
X
F
B
AX = 26
XD = 13
What is the relationship
between the length of AX
and XD?
Theorem 6-1
• The length of the segment from the vertex to
the centroid is twice the length of the
segment from the centroid to the midpoint.
2x
x
Example 3
Y
• Given
XYZ, find YQ if
QM = 4.
• ANSWER = 8
N
X
Q
R
M
Z
Example 4
Y
• Given
XYZ and if
QZ = 18, what is ZN?
• ANSWER = 27
N
X
Q
R
M
Z
Real-World Application:
Electrocardiogram Test
• When a health professional wants to test the
fitness of a person’s heart, they give an
electrocardiogram (EKG or ECG) test.
• An EKG test translates the heart’s electrical
activity into line tracings on a paper called an
electrocardiograph.
– The spikes and dips on the line tracings are called
waves
Real-World Application:
Electrocardiogram Test
• Electrodes are
placed on
body (chest
and both arms
and legs)to
record
electrical
activity of the
heart.
Real-World Application:
Electrocardiogram Test
You are a doctor. The cardiology technician
gives you the results of your patient’s EKG test:
– Equilateral triangle BCD is
used to plot the EKG
reading.
– The patient has a left
shoulder reading (S) of -1,
a right shoulder reading
(R) of 2, and a left leg
reading (L) of 3.
– You find the vertices of
Triangle SRL (Einthoven’s
triangle). Let’s construct
the centroid P in Algodoo!
Problem and graphic courtesy of McDougal Littell’s 2004 Geometry textbook
Screenshot of Algodoo Scene:
The purple bars represent the lines the students were
to use to designate the medians for Triangle SRL
Summary
• Median: A segment in which one endpoint is the vertex
of a triangle and the other is the midpoint of the side
of the vertex.
– Remember: A triangle has 3 medians.
• Centroid: The point of intersection of the three
medians of a triangle.
• Concurrent: Three or more lines or segments that meet
at a common point.
• Theorem 6-1: The length of the segment from the
vertex to the centroid is twice the length of the
segment from the centroid to the midpoint.
Class Work
CPS Clicker Questions
Questions adapted from Kuta Software problems on
Medians
(http://www.kutasoftware.com/FreeWorksheets/Geo
Worksheets/5-Medians.pdf)
In Triangle GFE GT is a median. Find TE
if FE = 8. G
A.
B.
C.
D.
16
10
4
8
[Default]
[MC Any]
[MC All]
F
T
E
In Triangle FGE, ET is a median. Find GF
if TF = 6.3.
A.
B.
C.
D.
6.3
12.6
3.15
5
F
T
G
[Default]
[MC Any]
[MC All]
E
In Triangle JIK, KL and IN are medians.
Find LJ if IJ = 6.
A.
B.
C.
D.
12
4
3
6
[Default]
[MC Any]
[MC All]
I
L
J
N
K
P is the centroid of triangle JKL and
PM = 8. Find JP.
A.
B.
C.
D.
16
4
12
8
[Default]
[MC Any]
[MC All]
J
N
L
P
M
K
P is the centroid of triangle QRS and
PT = 5. Find RT.
A.
B.
C.
D.
R
5
10
8
15
[Default]
[MC Any]
[MC All]
V
U
P
S
T
Q

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