Median is a line passing through the midpoint of each side of a triangle.
Let us see how to find the slope of median in a triangle.
Example :
A triangle has vertices at (6, 7), (2, -9) and (-4, 1). Find the slopes of its medians.
Solution :
Let A(6, 7), B(2, -9) and C(-4, 1) be the vertices of triangle.
Midpoint of the side BC = D
Midpoint of the side AC = E
Midpoint of the side AB = F
Medians of triangle are AD, BE and CF.
Slope of Median AD :
To find the slope median AD, find the point D which is the midpoint of B and D.
Midpoint formula = ((x_{1} + x_{2})/2, (y_{1} + y_{2})/2)
Midpoint of BC = ((2 - 4)/2, (-9 + 1)/2)
= (-2/2, -8/2)
= D(-1, -4)
Using the points A and D, find the slope of the median AD.
A(6, 7), D(-1, -4)
Slope of median AD = (y_{2} - y_{1})/(x_{2 }- x_{1})
= (-4 - 7)/(-1 - 6)
= -11/(-7)
= 11/7
Slope of Medina BE :
A(6, 7), C(-4, 1)
Midpoint of the side AC = ((6 -4)/2, (7 + 1)/2)
= (2/2, 8/2)
= E(1, 4)
Using the points B and E, find the slope of the median BE.
B(2, -9), E(1, 4)
Slope of median BE = (y_{2} - y_{1})/(x_{2 }- x_{1})
= (4 + 9)/(1 - 2)
= 13/(-1)
= -13
Slope of Medina CF :
A(6, 7), B(2, -9)
Midpoint of the side AC = (6 + 2)/2, (7 - 9)/2)
= (8/2, -2/2)
= F(4, -1)
Using the points C and F, find the slope of the median CF.
C(-4, 1), F(4, -1)
Slope of median CF = (y_{2} - y_{1})/(x_{2 }- x_{1})
= (-1 - 4)/(4 - 1)
= -5/3
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