Question

The sides of a triangle are equal and have equations 2x-y=0,3x+y=0 ,x-3y+10=0, respectively find the equation of three medians of the triangle and verify that they are concurrent

Answer 1

$2x-y=0,3x+y=0$ solving these equations we get $x=0,y=0$...A

$3x+y=0,x-3y=-10$ solving these equations we get $x=-1,y=3$...B

$x-3y=-10,2x-y=0$ solving these equations we get $x=2,y=4$...C

Let D be the mid point of line AB $\to D(-\frac{1}{2},\frac{3}{2})$

Let E be the mid point of line AC $\to E(1,2)$

Let F be the mid point of line BC $\to F(\frac{1}{2},\frac{7}{2})$

The required equations medians AF,CD,BE are: $\frac{y-y_1}{y_2-y_1}=\frac{x-x_1}{x_2-x_1}$

AF: $y=7x$

BE: $2y+x=5$

CD: $y=x+2$

condition for concurrency

$det\left(\begin{array}{ccc}7& -1& 0\\ 1& 2& 5\\ -1& 1& 2\end{array}\right)$

$=7\cdot det\left(\begin{array}{cc}2& 5\\ 1& 2\end{array}\right)-(-1)det\left(\begin{array}{cc}1& 5\\ -1& 2\end{array}\right)+0\cdot det\left(\begin{array}{cc}1& 2\\ -1& 1\end{array}\right)$

$7(-1)-(-1)\cdot 7+0\cdot 3$

$=0$

Therefore these lines are concurrent to each other