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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. Let's try this out with a particular triangle. Consider the triangle ABC with A=(5, 4) B=(-9, 6) C=(1, -4).

(a) Let D, E, F be the midpoints of  BC, AB, AC  respectively. Find the equations of medians  AD, BE, CF and 

(b) Show that the three medians in part (a) all pass through the same point.

 Mar 22, 2024
 #1
avatar+129659 
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Midpoint BC  =  [ (-9 + 1)/2 , (6 - 4) / 2 ] = [ -4, 1]

Midpoint AB  = [ (5 -9)/2 , (4 + 6) / 2 ] = [ -2, 5]   

Midpoint AC = [ (5 + 1) / 2 , (4 -4) / 2 ] = [ 3 , 0 ]   

 

Slope  AD  =  [ 4 - 1 ] / [ 5 - -4] =  1/3

Slope CE  = [ -4-5] / [ 1 - -2] =  -3

Slope BF =  [ 6-0] / [ -9-3]  = -1/2

 

Equation of line  through AD

y = (1/3)(x + 4) + 1                   (1)

Equation of line  through CE

y =  -3 ( x + 2) + 5              (2)

Equation of line  through BF

y = (-1/2)(x -3)       (3)

 

Set (1), (2)  equal  to find their x intersection

(1/3)(x + 4) + 1 =  -3(x + 2) + 5

x + 4 +  3  =  -9(x + 2) + 15

x+ 7 = -9x -18 + 15

x + 7 = -9x -3

10x = -10

x = -1

y = -3(-1+2) + 5 =  2

 

Intersection pt  =  ( -1 , 2)

 

Test in (3)

2  = (-1/2) (-1 -3) 

2 = 2         true  

 

cool cool cool

 Mar 23, 2024

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