# How do you find the coordinates of the other endpoint of a segment with the given endpoint is K(5,1) and the midpoint is M(1,4)?

May 23, 2018

See a 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:

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

Substituting the values from the points in the problem gives:

$\left(1 , 4\right) = \left(\frac{\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{1} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

We can now solve for $\textcolor{b l u e}{{x}_{2}}$ and $\textcolor{b l u e}{{y}_{2}}$

• $\textcolor{b l u e}{{x}_{2}}$:

$\frac{\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}}}{2} = 1$

$\textcolor{g r e e n}{2} \times \frac{\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}}}{2} = \textcolor{g r e e n}{2} \times 1$

$\cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}} = 2$

$\textcolor{red}{5} + \textcolor{b l u e}{{x}_{2}} = 2$

$\textcolor{red}{5} - \textcolor{g r e e n}{5} + \textcolor{b l u e}{{x}_{2}} = 2 - \textcolor{g r e e n}{5}$

$0 + \textcolor{b l u e}{{x}_{2}} = - 3$

$\textcolor{b l u e}{{x}_{2}} = - 3$

• $\textcolor{b l u e}{{y}_{2}}$

$\frac{\textcolor{red}{1} + \textcolor{b l u e}{{y}_{2}}}{2} = 4$

$\textcolor{g r e e n}{2} \times \frac{\textcolor{red}{1} + \textcolor{b l u e}{{y}_{2}}}{2} = \textcolor{g r e e n}{2} \times 4$

$\cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{1} + \textcolor{b l u e}{{y}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}} = 8$

$\textcolor{red}{1} + \textcolor{b l u e}{{y}_{2}} = 8$

$\textcolor{red}{1} - \textcolor{g r e e n}{1} + \textcolor{b l u e}{{y}_{2}} = 8 - \textcolor{g r e e n}{1}$

$0 + \textcolor{b l u e}{{y}_{2}} = 7$

$\textcolor{b l u e}{{y}_{2}} = 7$

The Other End Point Is: $\left(\textcolor{b l u e}{- 3} , \textcolor{b l u e}{7}\right)$