THE MIDPOINT FORMULA
LESSON READ-THROUGH
by Dr. Carol JVF Burns (website creator)
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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:

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
by practicing the exercise at the bottom of this page.


When you're done practicing, move on to:
Introduction to Equations and Inequalities in Two Variables

 
 
CONCEPT QUESTIONS EXERCISE:
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.
PROBLEM TYPES:
1 2 3 4 5
AVAILABLE MASTERED IN PROGRESS