Question

If the coordinates of the mid-points of the sides of a triangle are $(1,1),(2,-3)$ and $(3,4)$, find the vertices of the triangle.

Answer 1

From given, we have,

Let $A(x_1,y_1),B(x_2,y_2)$ has mid point $(1,1)$

$\therefore x_1+x_2=2,y_1+y_2=2$

Let $B(x_2,y_2),C(x_3,y_3)$ has mid point $(2,-3)$

$\therefore x_2+x_3=4,y_2+y_3=-6$

Let $C(x_3,y_3),A(x_1,y_1)$ has mid point $(3,4)$

$\therefore x_3+x_1=6,y_3+y_1=8$

solving the above equations, we get,

$x_1-x_3=-4,y_1-y_3=-6$

using all the 4 equations, we get,

$A(x_1,y_1)=(2,8)$

$B(x_2,y_2)=(4,-6)$

$C(x_3,y_3)=(0,0)$