audio read-through The Midpoint Formula

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THE MIDPOINT FORMULA
The midpoint of the line segment between points $\,(x_1,y_1)\,$ and $\,(x_2,y_2)\,$ is given by the Midpoint Formula: $$ \cssId{s6}{\left( \frac{x_1+x_2}2,\frac{y_1+y_2}2 \right)} $$

Here, $\,x_1\,$ (read as ‘$\,x\,$ sub $\,1\,$’) denotes the $\,x$-value of the first point, and $\,y_1\,$ (read as ‘$\,y\,$ sub $\,1\,$’) denotes the $\,y$-value of the first point. Similarly, $\,x_2\,$ and $\,y_2\,$ denote the $\,x$-value and $\,y$-value of the second point.

Thus, to find the location that is exactly halfway between two points, you average the $x$-values, and average the $y$-values.

The Midpoint Formula follows easily from the following observations:

proving the midpoint formula

Examples

Question: Find the midpoint of the line segment between $\,(1,-3)\,$ and $\,(-2,5)\,.$
Solution: $$ \cssId{s33}{\left( \frac{1+(-2)}2, \frac{-3 + 5}2 \right)} \cssId{s34}{= \left(-\frac12,1\right)} $$
Question: Suppose that $\,(2,3)\,$ is exactly halfway between $\,(-1,5)\,$ and $\,(x,y)\,.$ Find $\,x\,$ and $\,y\,.$
Solution: Rephrasing, $\,(2,3)\,$ is the midpoint of the segment with endpoints $\,(-1,5)\,$ and $\,(x,y)\,.$ Thus:
$\displaystyle \begin{align} &\cssId{s41}{(2,3)}\cr &\quad \cssId{s42}{= \left(\frac{-1+x}2,\frac{5+y}2\right)} \end{align} $ use the Midpoint Formula
$\displaystyle \begin{align} &\cssId{s44}{2 = \frac{-1+x}2}\cr &\quad \cssId{s45}{\text{and }}\ \cssId{s46}{3 = \frac{5+y}2} \end{align} $ for ordered pairs to be equal, the first coordinates must be equal and the second coordinates must be equal
$\displaystyle \begin{align} &\cssId{s50}{4 = -1 + x}\cr &\quad \cssId{s51}{\text{and }} \cssId{s52}{6 = 5 + y} \end{align} $ clear fractions (multiply both sides of both equations by $\,2$)
$5 = x\ $ and $\ 1 = y$ finish solving each equation
$x = 5\ $ and $\ y = 1$ write your solutions in the conventional way

Concept Practice