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# Help!

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A median

Nov 4, 2019
edited by tomsun  Nov 8, 2019

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Consider one specific triangle ABC with A=(4,8), B=(2,-6), and C=(-4,4).

The midpoint of  AB  =  [ (2 + 4) / 2 , ( 8  - 6) / 2  ]  =  (3, 1)

The slope of the line drawn between  C  and this point is  [ 4 - 1] / [ -4-3]  = 3/-7 = (-3/7)

So....the equation of the line joining these two points is

y  =(-3/7) ( x - 3) + 1

y = (-3/7)x + 9/7 + 1

y = (-3/7)x + 16/7       (1)

Likewise....the midpoint  of BC  = [ (-4 + 2) / 2,  (-6 + 4) / 2 ] =   ( -1, -1)

And the slope of the line joining A  and this point is   [ 8 - - 1 ] / [ 4 - - 1 ]  =  9/5

So....the line joining these two points is given by :

y  = (9/5)(x - 4) + 8

y = (9/5)x - 36/5 + 40/5

y = (9/5)x + 4/5      (2)

Setting (1)  and (2)  equal  we can find the x coordinate of the intersection point as

(-3/7)x + 16/7  =  (9/5)x + 4/5

16/7 - 4/5  =  (9/5 + 3/7)x

52/35  = 78/35x

52 = 78x

x = 52/78  =  2/3

And the y coordinate is  (9/5)(2/3) + 4/5  = 18/15 + 4/5   =  6/5 + 4/5  = 10/5   = 2

Here is a pic :    Nov 4, 2019