Geo2.36

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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\,


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