Geo2.36

Find the equations of the medians of the triangle formed by $(2,-5),(-3,4),(0,-3)\,$

The mid points of the three sides of the triangle with the three vertices given are

$(\frac{2-3}{2},\frac{-5+4}{2}),(\frac{-3+0}{2},\frac{4-3}{2}),(\frac{0+2}{2},\frac{-3-5}{2})\,$

$(\frac{-1}{2},\frac{-1}{2}),(\frac{-3}{2},\frac{1}{2}),(1,-4)\,$

Let the points be A,B and C.

The median AB is $y+\frac{1}{2}=\frac{1}{-1}(x+\frac{1}{2})\,$

$2y+1=-2x-1\,$

$2x+2y+2=0,x+y+1=0\,$

The second median BC is

$y-\frac{1}{2}=\frac{-4-\frac{1}{2}}{1+\frac{3}{2}}(x+\frac{3}{2})\,$

$2y-1=\frac{-7}{5}(2x+3)\,$

$10y-5=-14x-21,14x+10y+16=0,7x+5y+8=0\,$

The next median CA is

$y+4=\frac{\frac{-1}{2}+4}{\frac{-1}{2}-1}(x-1)\,$

$y+4=\frac{7}{-3}(x-1),-3y-12=7x-7,7x+3y+5=0\,$

Hence the medians are $x+y+1=0,7x+5y+8=0, 7x+3y+5=0\,$