# How do use midpoint formula to find an endpoint?

Oct 29, 2014

If you know one endpoint $\left({x}_{1} , {y}_{1}\right)$ and the midpoint $\left(a , b\right)$, but you do not know the other endpoint $\left({x}_{2} , {y}_{2}\right)$, then by rewriting the midpoint formula:

$\left\{\begin{matrix}a = \frac{{x}_{1} + {x}_{2}}{2} R i g h t a r r o w 2 a = {x}_{1} + {x}_{2} R i g h t a r r o w {x}_{2} = 2 a - {x}_{1} \\ b = \frac{{y}_{1} + {y}_{2}}{2} R i g h t a r r o w 2 b = {y}_{1} + {y}_{2} R i g h t a r r o w {y}_{2} = 2 b - {y}_{1}\end{matrix}\right.$

So, the unknown endpoint can be found by

$\left({x}_{2} , {y}_{2}\right) = \left(2 a - {x}_{1} , 2 b - {y}_{1}\right)$

I hope that this was helpful.