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Question

Find the equations of the medians of a triangle, the equations of whose sides are :
$$3 x + 2 y + 6 = 0,2 x - 5 y + 4 = 0$$  and  $$x - 3 y - 6 = 0.$$


Solution


The equations of the sides of a triangle are given by the following equations
$$3x+2y+6=0\quad \longrightarrow \left( 1 \right) $$
$$2x-5y+4=0\quad \longrightarrow \left( 2 \right) $$
$$x-3y-6=0\quad \longrightarrow \left( 3 \right) $$
Solving the equations taking two at a time we can find the vertices of the triangle whose sides are given by $$(1)$$, $$(2)$$ and $$(3)$$
Solving $$(1)$$ $ $$(2)$$
$$3x+2y+6=0$$ and $$2x-5y+4=0$$
we get, $$x=-2$$ and $$y=0$$
Solving $$(2)$$ & $$(3)$$
$$x-3y-6=0$$
$$2x-5y+4=0$$    } $$\Rightarrow$$ $$x=-42$$ and $$y=-16$$
Solving $$(3)$$ & $$(1)$$
$$x-3y-6=0$$
$$3x+2y+6=0$$    } $$\Rightarrow$$ $$x=\dfrac { -6 }{ 11 } $$ and $$y=\dfrac { -24 }{ 11 } $$
So, the vertices are
$$A(-2,0)$$  $$B=(-42,-16)$$  $$C=\left( \dfrac { -6 }{ 11 } ,\dfrac { -24 }{ 11 }  \right) $$
let the midpoint of the side $$AB$$ be $$P$$, which has coordinates $$(-22,-8)$$
let the midpoint of the side $$BA$$ be $$Q$$, which has coordinates $$\left( \dfrac { -468 }{ 11 } ,\dfrac { -200 }{ 11 }  \right) $$
let the midpoint of the side $$CA$$ be $$R$$, which has coordinates $$\left( \dfrac { -28 }{ 11 } ,\dfrac { -24 }{ 11 }  \right) $$
Now the equation of a line in the two-point form is given by 
$$y-{ y }_{ 1 }=\dfrac { { y }_{ 2 }-{ y }_{ 1 } }{ { x }_{ 2 }-{ x }_{ 1 } } \left( x-{ x }_{ 1 } \right) $$
So the equation of the median $$PC$$ is given by : $$y+\dfrac { 24 }{ 11 } =\dfrac { -8+\dfrac { 24 }{ 11 }  }{ -22+\dfrac { 6 }{ 11 }  } \left( x+\dfrac { 6 }{ 11 }  \right) $$
$$\Rightarrow 11y+24=\dfrac { 64 }{ 236 } \left( 11x+6 \right) $$
$$\Rightarrow 11y+24=\dfrac { 16 }{ 59 } \left( 11x+6 \right) $$
$$\Rightarrow \boxed { 176x-649y-1032=0 } $$
the equation of the median $$RB$$ is given by 
$$y+\dfrac { 24 }{ 11 } =\dfrac { -16+24/11 }{ -42+28/11 } \left( x+\dfrac { 28 }{ 11 }  \right) \Rightarrow 2387y+5208=836x+2128$$
                                                                 $$\Rightarrow \boxed { 836x-2387y-3080=0 } $$
the equation of the median $$AQ$$ is given by 
$$y-0=\dfrac { -200/11 }{ \dfrac { -468 }{ 11 } +2 } \left( x+2 \right) $$
$$\Rightarrow \boxed { 100x-223y+200=0 } $$

1217966_1304416_ans_50fd325d0b964d56b19f08f1102ba0ef.png

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