Distance Between Points; the Midpoint Formula
The Distance Formula
To find the distance between any two points in a coordinate plane:
- Subtract the $x$-values in any order; square the result.
- Subtract the $y$-values in any order; square the result.
- Add together the previous two quantities.
- Take the square root of the result.
This sequence of operations is expressed in the Distance Formula:
The distance between points $\,(x_1,y_1)\,$ and $\,(x_2,y_2)\,$ is given by the Distance Formula:
$$ \cssId{s10}{\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}} $$For a complete review of the distance between two points in the coordinate plane, study the following lesson, which includes a derivation, a discussion of subscript notation, and examples: The Distance Formula.
The Midpoint Formula
To find the midpoint of the line segment between any two points in a coordinate plane:
- Average the $x$-values of the two points—that is, add them and divide by $2\,.$ This gives the $x$-value of the midpoint.
- Average the $y$-values of the two points—that is, add them and divide by $2\,.$ This gives the $y$-value of the midpoint.
This sequence of operations is expressed in 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:
$$ \cssId{s25}{\left( \frac{x_1+x_2}2,\frac{y_1+y_2}2 \right)} $$For a complete review of midpoints, including a derivation and examples, study: The Midpoint Formula.