# In the standard (x,y) coordinate plane, point M with coordinates (5,4) is the midpoint of AB, and B has coordinates (7,3). What are the coordinates of A ?

Jul 27, 2016

Coordinates of A are $\left(3 , 5\right)$

#### Explanation:

In a segment if $A \left({x}_{1} , {y}_{1}\right)$ and $B \left({x}_{2} , {y}_{2}\right)$ are coordinates of its end points, the coordinates of midpoint $M$ are $\frac{{x}_{1} + {x}_{2}}{2} , \frac{{y}_{1} + {y}_{2}}{2}$.

Now we are given $A \left({x}_{1} , {y}_{1}\right)$ and $B \left(7 , 3\right)$, hence midpoint is $\frac{{x}_{1} + 7}{2} , \frac{{y}_{1} + 3}{2}$, but midpoint is given to be $\left(5 , 4\right)$.

Hence $\frac{{x}_{1} + 7}{2} = 5$ i.e. ${x}_{1} + 7 = 5 \times 2 = 10$ and ${x}_{1} = 10 - 7 = 3$

and $\frac{{y}_{1} + 3}{2} = 4$ i.e. ${y}_{1} + 3 = 4 \times 2 = 8$ and ${x}_{1} = 8 - 3 = 5$

Hence, coordinates of A are $\left(3 , 5\right)$