# How does one derive the Midpoint Formula?

Nov 23, 2015

It can be prooven using vectors. See explanation.

#### Explanation:

Let there be 2 points: A=(x_A;y_A) and $B = \left({x}_{B} , {y}_{B}\right)$. We are looking for a point $M$ for which vectors $\vec{A M}$ and $\vec{M B}$ are equal. Using the equality of vectors we have:

[x_M-x_A;y_M-y_A]=[x_B-x_M;y_B-y_M].

Now we can calculate both coordinates separately:

${x}_{M} - {x}_{A} = {x}_{B} - {x}_{M}$

${x}_{M} + {x}_{M} = {x}_{B} + {x}_{A}$

$2 {x}_{M} = {x}_{A} + {x}_{B}$

${x}_{M} = \frac{{x}_{A} + {x}_{B}}{2}$

For $y$ coordinate we have similar equation:

${y}_{M} - {y}_{A} = {y}_{B} - {y}_{M}$

${y}_{M} + {y}_{M} = {y}_{B} + {y}_{A}$

$2 {y}_{M} = {y}_{A} + {y}_{B}$

${y}_{M} = \frac{{y}_{A} + {y}_{B}}{2}$