What Is The Distance Formula? How To Calculate Distance Between Two Points?


What is The Distance Formula?

If you wanted to calculate the distance between two points then you can use the distance formula in that case. This formula has been derived from the Pythagorean Theorem and easy to implement as well. In simple words, we can say that Distance Formula is a variant of Pythagorean Theorem used back in the geometry.

How To Calculate Distance Between Two Points?

For example, when you have to find the distance between two points (–2, 1) and (1, 5) then it will make a straight line in the form of right-angle triangle where these points will be fixed at two corners and makes it easy to find the horizontal and vertical lines of a triangle. Then you can apply the Pythagorean theorem finally to find the third side or hypotenuse of a triangle. And the height of hypotenuse is actually the distance between two points. Let us see how this theory can be written in the form of a formula.

The Distance Formula

When two points (x1, y1) and (x2, y2) are given then the distance d between these points can becalculated by the formula:

\[\large d=\sqrt{\left(x_{2}^{2}-x_{1}^{2}\right)+\left(y_{2}^{2}-y_{1}^{2}\right)}\]

Don’t scare off these subscripts because they are just indicating the first and second point on a straight line. Two mean the line has two points and you have to calculate the distance between these points only.  Distance can be calculated in terms of speed and time as well where distance is given by speed/time and frequently used for mathematics or physics calculations. You can use any of them based on the nature of a given problem.

Example For The Distance Formula

Question: Given the points (1, -2) and (-3, 5), find the distance between them.

Solution:

Label the points as follows

\[\ \left(x_{1},y_{1}\right)=\left(-1,-2\right) and \left(x_{2},y_{2}\right)=\left(-3,-5\right)\]

\[\ Therefore\; x_{1}=-1,\: y_{1}=-2,\: x_{2}=-3, and\: y_{2}=5\]

To find the distance (d) between the points, use the distance formula:

\[\ d=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}\]

\[\ =\sqrt{\left(-3-\left(-1\right)\right)^{2}+\left(5-\left(-2\right)\right)^{2}}\]

\[\ =\sqrt{\left(-3+1\right)^{2}+\left(5+2\right)^{2}}\]

\[\ =\sqrt{\left(-2\right)^{2}+\left(7\right)^{2}}\]

\[\ \sqrt{4+49} \]

\[\ =\sqrt{53}\]


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