# How do you find the coordinates of the other endpoint of a segment with the given P = (5, 6) and midpoint M = (8, 2)?

Apr 5, 2017

See the entire solution process below:

#### Explanation:

The formula to find the mid-point of a line segment give the two end points is:

$M = \left(\frac{\textcolor{red}{{x}_{1}} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{{y}_{1}} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

Where $M$ is the midpoint and the given points are:

$\left(\textcolor{red}{\left({x}_{1} , {y}_{1}\right)}\right)$ and $\left(\textcolor{b l u e}{\left({x}_{2} , {y}_{2}\right)}\right)$

Substituting the information we have from the problem gives:

$\left(8 , 2\right) = \left(\frac{\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{6} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

First, we can solve for ${x}_{2}$:

$8 = \frac{5 + {x}_{2}}{2}$

$\textcolor{red}{2} \times 8 = \textcolor{red}{2} \times \frac{5 + {x}_{2}}{2}$

$16 = \cancel{\textcolor{red}{2}} \times \frac{5 + {x}_{2}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}$

$16 = 5 + {x}_{2}$

$- \textcolor{red}{5} + 16 = - \textcolor{red}{5} + 5 + {x}_{2}$

$11 = 0 + {x}_{2}$

$11 = {x}_{2}$

Next, we can solve for ${y}_{2}$

$2 = \frac{6 + {y}_{2}}{2}$

$\textcolor{red}{2} \times 2 = \textcolor{red}{2} \times \frac{6 + {y}_{2}}{2}$

$4 = \cancel{\textcolor{red}{2}} \times \frac{6 + {y}_{2}}{\textcolor{red}{\cancel{\textcolor{b l a c k}{2}}}}$

$- \textcolor{red}{6} + 4 = - \textcolor{red}{6} + 6 + {y}_{2}$

$- 2 = 0 + {y}_{2}$

$- 2 = {y}_{2}$

The other end point is $\left(11 , - 2\right)$