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6-1 Medians
Objective: To identify and construct medians in triangles
February 2011Fellow: Brooke OdleTeacher: Ms. SanchezSaint Vincent Academy
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Lesson Overview
• Introduce topic– Vocabulary and Theorem 6-1
• Example Problems• Real World Application– Electrocardiograms
• Class work– Clicker Questions
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What is a median?
• 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.
B
A
C
D
median BD
median TU
U
S
R
T
median ZW
YW
X
Z
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Example 1
• In ABC, CE and AD are medians.
• Find BE if AB = 18.• ANSWER: 9
A
B
C
D
E
18
?
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Example 2
• In ABC, CE and AD are medians.
• If CD = 2x+5, BD = 4x-1, and AE = 5x-2, find BE.
• ANSWER: 13
A
B
C
D
E
5x-2
?
2x+5
4x-1
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Intersections of Medians• The medians of triangle
JKM, JR, KP, and MQ, intersect at a common point called the centroid.
• When 3 or more lines or segments meet at the same point, the lines are concurrent.
M
P
J
QK
X
X is the centroid of JKM and JR, KP, and MQ are all concurrent.
R
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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.
FX
B
A
E
C
2613
AX = 26XD = 13
What is the relationship between the length of AX and XD?
D
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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
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Example 3
• Given XYZ, find YQ if QM = 4.
• ANSWER = 8 N
M
R
Y
Z
QX
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Example 4
• Given XYZ and if QZ = 18, what is ZN?
• ANSWER = 27 N
M
R
Y
Z
QX
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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
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Real-World Application:Electrocardiogram Test
• Electrodes are placed on body (chest and both arms and legs)to record electrical activity of the heart.
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Real-World Application:Electrocardiogram 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!
You are a doctor. The cardiology technician gives you the results of your patient’s EKG test:
Problem and graphic courtesy of McDougal Littell’s 2004 Geometry textbook
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Screenshot of Algodoo Scene:The purple bars represent the lines the students were to use to designate the medians for Triangle SRL
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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.
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CPS Clicker QuestionsQuestions adapted from Kuta Software problems on
Medians (http://www.kutasoftware.com/FreeWorksheets/
GeoWorksheets/5-Medians.pdf)
Class Work
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In Triangle GFE GT is a median. Find TE if FE = 8.
A. 16B. 10C. 4D. 8
[Default][MC Any][MC All]
F T E
G
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In Triangle FGE, ET is a median. Find GF if TF = 6.3.
A. 6.3B. 12.6C. 3.15D. 5
[Default][MC Any][MC All]
E
G
F T
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In Triangle JIK, KL and IN are medians. Find LJ if IJ = 6.
A. 12B. 4C. 3D. 6
[Default][MC Any][MC All]
I
L
J KN
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P is the centroid of triangle JKL and PM = 8. Find JP.
A. 16B. 4C. 12D. 8
[Default][MC Any][MC All]
K
M
L
N
J
P
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P is the centroid of triangle QRS and PT = 5. Find RT.
A. 5B. 10C. 8D. 15
[Default][MC Any][MC All]
Q
R
SP
T
U
V