The distance and midpoint formulas
The distance formula is used to find the distance between two points in the coordinate plane. We'll explain this using an example below
We want to calculate the distance between the two points (-2, 1) and (4, 3). We could see the line drawn between these two points is the hypotenuse of a right triangle. The legs of this triangle would be parallel to the axes which mean that we can measure the length of the legs easily.
We'll get the length of the distance d by using the Pythagorean Theorem
$$d^{2}=2^{2}+6^{2}=4+36=40$$
$$d=\sqrt{40}\approx 6.32$$
This method can be used to determine the distance between any two points in a coordinate plane and is summarized in the distance formula
$$d=\sqrt{\left ( x_{2}-x_{1} \right )^{2}+\left ( y_{2}-y_{1} \right )^{2}}$$
The point that is at the same distance from two points A (x1, y1) and B (x2, y2) on a line is called the midpoint. You calculate the midpoint using the midpoint formula
$$m =\left ( \frac{x_{1}+x_{2}}{2} \right ),\: \: \left ( \frac{y_{1}+y_{2}}{2} \right )$$
The midpoint formula works by taking the average of the x-coordinates and the average of the y-coordinates of the two endpoints. This gives you the point that lies exactly halfway between them.
We can use the example above to illustrate this
$$ m =\left ( \frac{4+(-2)}{2} \right ),\: \: \left ( \frac{3+1}{2} \right )=$$
$$=\left ( \frac{2}{2} \right ),\: \: \left ( \frac{4}{2} \right )=\begin{pmatrix} 1,\: 2 \end{pmatrix}$$
Let's look at one more example, this time with negative coordinates. Find the midpoint between the points (-5, 3) and (1, -7).
$$ m =\left ( \frac{-5+1}{2} \right ),\: \: \left ( \frac{3+(-7)}{2} \right )=$$
$$=\left ( \frac{-4}{2} \right ),\: \: \left ( \frac{-4}{2} \right )=\begin{pmatrix} -2,\: -2 \end{pmatrix}$$
The midpoint of the line segment is (-2, -2).
Video lesson
Calculate the distance between the two points
