Problem on the formula for midpoints

$\newcommand{\bfA}{\mathbf{A}}$ $\newcommand{\bfB}{\mathbf{B}}$ $\newcommand{\bfC}{\mathbf{C}}$ $\newcommand{\bfF}{\mathbf{F}}$ $\newcommand{\bfI}{\mathbf{I}}$ $\newcommand{\bfa}{\mathbf{a}}$ $\newcommand{\bfb}{\mathbf{b}}$ $\newcommand{\bfc}{\mathbf{c}}$ $\newcommand{\bfd}{\mathbf{d}}$ $\newcommand{\bfe}{\mathbf{e}}$ $\newcommand{\bfi}{\mathbf{i}}$ $\newcommand{\bfj}{\mathbf{j}}$ $\newcommand{\bfk}{\mathbf{k}}$ $\newcommand{\bfn}{\mathbf{n}}$ $\newcommand{\bfr}{\mathbf{r}}$ $\newcommand{\bfu}{\mathbf{u}}$ $\newcommand{\bfv}{\mathbf{v}}$ $\newcommand{\bfw}{\mathbf{w}}$ $\newcommand{\bfx}{\mathbf{x}}$ $\newcommand{\bfy}{\mathbf{y}}$ $\newcommand{\bfz}{\mathbf{z}}$
Use vector addition, subtraction, and scalar multiplication to show that the midpoint between the two points $\mathbf{x}$ and $\mathbf{y}$ is $\frac{\mathbf{x} + \mathbf{y}}{2}$.

  • Solution

    Let's call the midpoint $\bfz$. Our goal is to relate $\bfz$ to $\bfx$ and $\bfy$.
    We choose to represent all line segments in this problem as vectors.
    Now, we interpret the fact that $\bfz$ is a midpoint in terms of vectors.
    Recall that
    The vector going from $\bfx$ to $\bfy$ is $\bfy - \bfx$.
    Similarly, the vector going from $\bfx$ to $\bfz$ is $\bfz - \bfx$.
    Because $\bfz$ is the midpoint of $\bfx\bfy$, the vector going from $\bfx$ to $\bfz$ has the same length and direction as the vector going from $\bfz$ to $\bfy$.
    Hence, we can write that $$ \bfz - \bfx = \bfy - \bfz.$$
    Visually, this equality can be depicted as:
    Adding $\bfz$ and $\bfx$ to both sides, we get \begin{align}
    2\bfz &= \bfx + \bfy.
    \end{align}
    After dividing by 2, we conclude $$ \bfz = \frac{\bfx + \bfy }{2}.$$

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