# How do you find the missing coordinate if P=(4,-1) is the midpoint of the segment AB, where A=(2, 5)?

Jun 15, 2016

$B = \left(6 , - 7\right)$

#### Explanation:

Recall that the midpoint formula is:

$\textcolor{b l u e}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} M = \left(\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}\right) \textcolor{w h i t e}{\frac{a}{a}} |}}}$

Let $M = \left(4 , - 1\right)$
Let $\left({x}_{1} , {y}_{1}\right) = \left(2 , 5\right)$
Let $\left({x}_{2} , {y}_{2}\right) =$coordinate of B

Start by plugging your known values into the formula.

$\left(4 , - 1\right) = \left(\frac{2 + {x}_{2}}{2} , \frac{5 + {y}_{2}}{2}\right)$

Since you are looking for $\left({x}_{2} , {y}_{2}\right)$, the coordinates of B, you can treat the components of $x$ and $y$ to be two separate equations. For instance,

$4 = \frac{2 + {x}_{2}}{2} \textcolor{w h i t e}{X X X X X X X X} - 1 = \frac{5 + {y}_{2}}{2}$

In each equation, solve for the variable.

$8 = 2 + {x}_{2} \textcolor{w h i t e}{X X X X X X X \times} - 2 = 5 + {y}_{2}$

${x}_{2} = 6 \textcolor{w h i t e}{X X X X X X X X X X X x} {y}_{2} = - 7$

Hence, the coordinate of $B$ is:

$B = \textcolor{g r e e n}{| \overline{\underline{\textcolor{w h i t e}{\frac{a}{a}} \textcolor{b l a c k}{\left(\left(6 , - 7\right)\right)} \textcolor{w h i t e}{\frac{a}{a}} |}}}$