Median Lines and the Centroid
In previous mathematics courses you investigated properties of geometric shapes. You also found equations of lines drawn in a coordinate plane. Analytic geometry expands the study of geometric shapes using equation of lines.
What information do you need to find the equation of a line?
slope and y­ intercept or the x and y coordinates of any two points on the line.
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Definition
Median Line: A line that is drawn from a vertex of a triangle to the midpoint of the opposite.
A
B
M
C
What would we need to know to find the median line from A? The midpoint of BC.
The slope of AM.
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Centroid: The centre of an object's mass; the point at which an object balances; the centroid is also known as the centre of gravity.
We can find the centroid of a triangle by finding the point of intersection of two median lines.
Steps:
1. Find the equation of one median line.
2. Find the equation of a second median line.
3. Find the point of intersection using the two equations from steps 1 and 2.
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Example: The coordinates of ∆ABC are A﴾4, 7﴿, B﴾­2, 3﴿ and C﴾6, ­1﴿. Find the intersection of the medians ﴾centroid﴿. 4
Same question choosing two other medians.
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Step 1: Find the median line from vertex A.
Midpoint of BC:
D
Slope of AD:
Equation of AD:
y = 3x ­ 5 6
Step 2: Find the median line from vertex C.
Midpoint of AB:
E
Slope of EC:
D
Equation of EC:
7
Step 3: Find the point of intersection.
E
P
D
the centroid is located at ﴾8/3, 3﴿
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The coordinates of ∆PQR are P (5, 5), Q (5, 10), and R (10, 8). Determine the coordinates of the centroid
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Homework:
text p. 79 #7, 12, 15
p. 120 #7, 10, 14
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