Expressing distance between any two points on the plane in the terms of their coordinates is defined distance between two points in terms of coordinates.

Geometrically, every point has its own coordinates on a plane as per the Cartesian coordinate system. The distance between any two points can be calculated by the coordinates of the two points. Developing a general mathematical expression which calculates the distance between the points by the coordinates is really useful to evaluate the distance between them easily.

Consider a point on a plane and assume to call it as point $P$. Assume the distance of the point $P$ from origin of the Cartesian coordinate system are ${x}_{1}$ and ${y}_{1}$. Therefore, the location of the point $P$ is $P({x}_{1},{y}_{1})$. Similarly, consider another point on the plane and assume to call it as point $Q$. Assume, the location of the point $Q$ in coordinates form is $Q({x}_{2},{y}_{2})$. There is some distance between points $P$ and $Q$ in the plane and expressing it in coordinates form is the object here.

Draw a line from point $P$ and it should be parallel to the $x$-axis. Similarly, draw another line from point $Q$ and it should be parallel to $y$-axis and perpendicular to the $x$-axis. Assume, the line drawn from point $P$ and the line drawn from point $Q$ are met at a point $R$. This process formed a right angled triangle $\Delta QPR$.

As per the right angled triangle $\Delta QPR$, the distance between two points is hypotenuse of the right angle triangle.

Hypotenuse of the right angled triangle can be calculated by applying Pythagorean theorem.

Pythagorean Theorem actually reveals the relation of the hypotenuse of the right angled triangle with opposite side and adjacent side. Therefore, let us first calculate the length of the opposite side and length of the adjacent side.

Length of Opposite side $(QR)=OQ\u2013OR={y}_{2}\u2013{y}_{1}$

Length of Adjacent side $(PR)=OR\u2013OP={x}_{2}\u2013{x}_{1}$

According to Pythagoras’ theorem,

${(Hypotenuse)}^{2}={(Opposite\; Side)}^{2}+{(Adjacent\; Side)}^{2}$

$\Rightarrow {PQ}^{2}={QR}^{2}+{PR}^{2}$

Therefore, distance between two points $PQ=\sqrt{{QR}^{2}+{PR}^{2}}$

$\Rightarrow PQ=\sqrt{{({y}_{2}\u2013{y}_{1})}^{2}+{({x}_{2}\u2013{x}_{1})}^{2}}$

It can also be written as follows

$\Rightarrow PQ=\sqrt{{({x}_{2}\u2013{x}_{1})}^{2}+{({y}_{2}\u2013{y}_{1})}^{2}}$

It is an expression which evaluates the distance between any two points on the plane with the help of the coordinates of the points.

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