Question

If a vertex of a triangle be $(1,1)$ and the middle points of the sides through it be $(-2,3)$ and $(5,2)$, find the other vertices.

Answer 1

Let $\Delta ABC$ be the triangle, $A(1,1)$, $B(x,y)$, $C(a,b)$

We have mid point formula,

$P(X,Y)=(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2})$

mid point of AC

$-2=\frac{1+a_1}{2}a=-53=\frac{1+b}{2}b=5$

mid point of AB

$\Rightarrow$ $5=\frac{x+1}{2}$

$\Rightarrow$ $x=9$

$\Rightarrow$ $\frac{1+y}{2}=2$

$\Rightarrow$ $y=3$

Other Vertices are$(9,3),(-5,5)$