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A median of a triangle is a line segment joining a vertex of a triangle to the midpoint of the opposite side. The three medians of a triangle are drawn below. Note that the three medians appear to intersect at the same point! Let's try this out with a particular triangle. Consider the triangle ABC with A = (3,6), B = (-5,2), and C = (7,-8). (a) Let D, E, F be the midpoints of \overline{BC}, \overline{AC}, \overline{AB}, respectively. Find the equations of medians \overline{AD}, \overline{BE}, and \overline{CF}. (b) Show that the three medians in part (a) all pass through the same point.

https://latex.artofproblemsolving.com/b/c/3/bc36d6a86ebdd87399c8b07d66ec053e8593c264.png

Mar 3, 2020

#1
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Hard to  see this....but

Midpoint of  AB =  [ (3 -5)/2,  (6 + 2) / 2  ]  =   [ -1, 4 ]  =  F

Midpoint of  of  BC  =  [ (-5 + 7)/2 , (2- 8)/2 ] =  [ 1 , -3]   =  D

Midpoint  of  AC   =   [ (3 + 7)/2 , ( 6 - 8)/2 ] =  [ 5, -1]  =  E

Slope of  CF =   [ -8 - 4 ]  / [ 7 - -1] =  -12/8 =  -3/2

Equation of  line  containing  CF =

y= -(3/2) ( x -  - 1)  + 4

y = -(3/2) ( x + 1)  + 4

y= (3/2)x - 3/2 + 4

y = -(3/2)x  +  5/2       (1)

Slope of AD  = [ 6 - -3 ] / [ 3 -  1] =  9/ 2

Equation of  line  containing AD  =

y = (9/2) (x -3) + 6

y =(9/2)x - 27/2  + 12/2

y= (9/2)x - 15/2      (2)

We  can  the  find  x intersection  of  medians  CF  and AD   by setting (1)  = (2)....so we have

-(3/2)x + 5/2 =  (9/2)x  - 15/2

10 = 6x

5 = 3x

x = 5/3

And using  (1)  the y value of the intersections is

y = -(3/2)(5/3) + 5/2  =  0

So....the intersection of the  medians  =  (5/3 ,  0)

We can check  this  by  writing an equation for  the  remaining median,  BE

Slope of line  containing  BE  =

[ 2 - -1 ]  / [ -5 - 5] =  3 / -10  =  -3/10

So the   equation of this line is

y = -(3/10)(x - - 5) + 2

y = - (3/10)x - 15/10  + 20/10

y= -(3/10)x +  5/10

y = -(3/10)x + 1/2

Note that when  x = (5/3)  we have that

y = -(3/10)(5/3) + 1/2

y =  - (5/10)  +  1/2

y=  -1/2 +  1/2

y  =  0

So.....this confirms that the  intersction of the  medians = ( 5/3, 0 )   Mar 4, 2020
#3
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OK....no prob  !!!   CPhill  Mar 4, 2020