Given the vertices of triangle ABC, A(1,2), B(2,3), C(2,-5), determine the median AE?
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The median AE is the line joining the point A(1,2) to the point between the points B(2, 3) and C(2 , -5).
Note: I am providing the equation of the median AE which I believe is what you want.
The coordinates of the pint between two points (x1, y1) and (x2, y2) is [( x1 + x2)/2 , (y1 + y2)/2]
The mid point E between B and C is [( 2+2)/2, (3 - 5)/2] = (2 , -1)
The equation of the line joining A and E is:
y - 2 = [( -1 - 2)/(2 - 1)]( x - 1)
=> y - 2 = -3*(x - 1)
=> y - 2 = -3x + 3
=> 3x + y - 5 = 0
The median AE is 3x + y - 5 = 0
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Since the median joins a vertex of triangle and the midpoint of the opposite side, we'll choose E as being the midpoint of the opposite side BC.
The coordinates of E are:
xE = (xB+xC)/2
xE = (2+2)/2
xE = 2
yE = (yB+yC)/2
yE = (3-5)/2
yE = -1
Now, the problem doesn't specify if we have to find the length of the median AE or the equation of the median AE.
In order to write the equation of the line AE, we have to know also the slope:
mAE=(yE-yA)/(xE-xA)
mAE= (-1-2)/(2-1)
mAE = -3
The equation of median AE is:
y-yA=mAE*(x-xA)
y-2=(-3)*(x-1)
y-2+3x-3=0
y+3x-5=0
On the other hand, the length of the segment AE is:
AE = sqrt[(xE-xA)^2 + (yE - yA)^2]
AE = sqrt[(2-1)^2 + (-1 - 2)^2]
AE = sqrt(1+9)
The length of the median AE is: AE = sqrt 10 units.
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