﻿ the Midpoint Formula
THE MIDPOINT FORMULA
by Dr. Carol JVF Burns (website creator)
Follow along with the highlighted text while you listen!

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:

• Referring to the sketch below, $\Delta ABD\,$ is similar to $\,\Delta AMC$.
That is, these two triangles have the same angles.
Why? They both share angle $\,A\,$, and they both have a right angle.
Since all the angles in a triangle sum to $\,180^\circ\,$, the third angles must also be the same.
• Similarity gives us what we need!
It tells us that $\Delta ABD\,$ and $\Delta AMC\,$ have exactly the same shapes—they're just different sizes.
Since $\,\overline{AM}\,$ is exactly half of $\,\overline{AB}\,$, $\,\overline{AC}\,$ must be exactly half of $\,\overline{AD}\,$.
Thus, $\,C\,$ is the midpoint between $\,A\,$ and $\,D\,$ (which can be found by averaging $\,x_1\,$ and $\,x_2\,$).
• Use a similar argument to show that $\,\overline{DE}\,$ (which has the same length as $\,\overline{CM}\,$) is exactly half of $\,\overline{DB}\,$.

EXAMPLES:
Question:
Find the midpoint of the line segment between $\,(1,-3)\,$ and $\,(-2,5)\,$.
Solution:
$\displaystyle \cssId{s33}{\left( \frac{1+(-2)}2, \frac{-3 + 5}2 \right)} \cssId{s34}{= \left(-\frac12,1\right)}$
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:
 $\displaystyle \cssId{s41}{(2,3)} \cssId{s42}{= \left(\frac{-1+x}2,\frac{5+y}2\right)}$ use the Midpoint Formula $\displaystyle 2 = \frac{-1+x}2\$ and $\ \displaystyle 3 = \frac{5+y}2$ for ordered pairs to be equal, the first coordinates must be equal and the second coordinates must be equal $4 = -1 + x\$ and $\ 6 = 5 + y$ 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
Master the ideas from this section