## Problem on the formula for midpoints

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
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 $$\bfz = \frac{\bfx + \bfy }{2}.$$