# How do you find the missing coordinate if the 1st coordinate is C (6,5) has midpoint M (4,2)?

Aug 13, 2017

See a solution process below:

#### Explanation:

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

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

Where $\left({x}_{M} , {y}_{M}\right)$ is the midpoint and the given points are:

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

First, find the $x$ value of the missing coordinate:

Substituting the information from the problem gives:

$4 = \frac{\textcolor{red}{6} + \textcolor{b l u e}{{x}_{2}}}{2}$

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

$\textcolor{g r e e n}{2} \times 4 = \textcolor{g r e e n}{2} \left(\frac{\textcolor{red}{6} + \textcolor{b l u e}{{x}_{2}}}{2}\right)$

$8 = \cancel{\textcolor{g r e e n}{2}} \left(\frac{\textcolor{red}{6} + \textcolor{b l u e}{{x}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}\right)$

$8 = \textcolor{red}{6} + \textcolor{b l u e}{{x}_{2}}$

$- \textcolor{g r e e n}{6} + 8 = - \textcolor{g r e e n}{6} + \textcolor{red}{6} + \textcolor{b l u e}{{x}_{2}}$

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

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

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

First, find the $y$ value of the missing coordinate:

Substituting the information from the problem gives:

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

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

$\textcolor{g r e e n}{2} \times 2 = \textcolor{g r e e n}{2} \left(\frac{\textcolor{red}{5} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

$4 = \cancel{\textcolor{g r e e n}{2}} \left(\frac{\textcolor{red}{5} + \textcolor{b l u e}{{y}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}\right)$

$4 = \textcolor{red}{5} + \textcolor{b l u e}{{y}_{2}}$

$- \textcolor{g r e e n}{5} + 4 = - \textcolor{g r e e n}{5} + \textcolor{red}{5} + \textcolor{b l u e}{{y}_{2}}$

$- 1 = 0 + \textcolor{b l u e}{{y}_{2}}$

$- 1 = \textcolor{b l u e}{{y}_{2}}$

$\textcolor{b l u e}{{y}_{2}} = - 1$

The coordinates of the Midpoint are: $\left(2 , - 1\right)$