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A median of a triangle is a line segment from a vertex of a triangle to the midpoint of the opposite side of the triangle. Below, the three medians of the black triangle are shown in red.

 

 

Notice that the three medians appear to pass through the same point! Let's test this out with a specific triangle. Consider one specific triangle ABC with A=(4,8), B=(2,-6), and C=(-14,8).

 

I think I'm supposed to compute the equations of the medians, but I don't know how to do that.

 Feb 14, 2022
 #1
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OK

 

Let C = (-14,8)   and A  =  (4,8)

The midpoint of this  segment is   [  ( 4+ - 14) / 2  ,  ( 8 + 8)/2 ]  =   (-5, 8)

And this  median will pass through B =   (2, -6)

The slope of this median line   =  (-6-8)/(2- - 5)  = -14/7 =- 2

Using the midpoint we can write the equations of the line as

(y - 8) = -2 ( x - -5)

y = -2x -10 + 8

y = -2x -2

 

Let the midpoint between A and B be [ (4+2 ) /2 ,   (8-6)/2  ] = ( 3, 1)   

And this median will pass through C = ( -14,8)

The slope of this median line is  ( 1-8) / ( 3 - -14)  = -  7/17

And using the midpoint the equation of this median line is

y -1 = - (7/17) ( x - 3)

y = - (7/17)x + 21/17  + 1

y = -(7/17)x + 21/17 + 17/17

y = - (7/17)x  + 38/17

 

Let the midpoint of  B and C  be  [ ( -14+2)/2 , (-6+8)/2 ]  = (-6, 1)

This median will pass through A = ( 4,8)

The slope of this median =   (1-8)  / ( -6-4)   =  -7 / -10   =  7/10

Using the median the equation of this median line is

y - 8 =  (7/10)(x  -4)

y = (7/10)x - 28/10  + 8

y = (7/10)x -28/10 + 80/10

y = (7/10)x + 52/10  =  .7x  + 5.2

 

If we set any two of the lines =, we can find the x coordinate of the intersection of the medians

 

-2x - 2  =  .7x  + 5.2

-2 -5.2  = 2x + .7x

-7.2 = 2.7x

x = -7.2 / 2.7   =  -72/27  =  -8/3

 

And the y coordinate of the intersection is

 

-2(-8/3) - 2  =  16/3  - 2  =  16/3 - 9/3 =  7/3

 

 

 

cool cool cool

 Feb 14, 2022

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