# How do you find the coordinates of the other endpoint of a segment with the given endpoint T(-3.5,-6) and the midpoint M(1.5,4.5)?

May 20, 2017

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:

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

Substituting the information we have gives:

$\left(1.5 , 4.5\right) = \left(\frac{\textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}}{2} , \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}}{2}\right)$

To find $\textcolor{b l u e}{{x}_{2}}$ we need to solve this equation:

$1.5 = \frac{\textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}}{2}$

$\textcolor{g r e e n}{2} \times 1.5 = \textcolor{g r e e n}{2} \times \frac{\textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}}{2}$

$3 = \cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}$

$3 = \textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}$

$3.5 + 3 = 3.5 \textcolor{red}{- 3.5} + \textcolor{b l u e}{{x}_{2}}$

$6.5 = 0 + \textcolor{b l u e}{{x}_{2}}$

$6.5 = \textcolor{b l u e}{{x}_{2}}$

$\textcolor{b l u e}{{x}_{2}} = 6.5$

To find $\textcolor{b l u e}{{y}_{2}}$ we need to solve this equation:

$4.5 = \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}}{2}$

$\textcolor{g r e e n}{2} \times 4.5 = \textcolor{g r e e n}{2} \times \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}}{2}$

$9 = \cancel{\textcolor{g r e e n}{2}} \times \frac{\textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}}{\textcolor{g r e e n}{\cancel{\textcolor{b l a c k}{2}}}}$

$9 = \textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}$

$9 + 6 = 6 \textcolor{red}{- 6} + \textcolor{b l u e}{{y}_{2}}$

$15 = 0 + \textcolor{b l u e}{{y}_{2}}$

$15 = \textcolor{b l u e}{{y}_{2}}$

$\textcolor{b l u e}{{y}_{2}} = 15$

The other end point of the segment is: $\left(\textcolor{b l u e}{6.5} , \textcolor{b l u e}{15}\right)$