# Geometry: Distance Formula

Usually the coordinate place consists of two axes namely X axis and Y axis. Thus, a point in a coordinate plane is usually denoted by the coordinates (x,y) where x is the perpendicular distance of the point from Y-axis and y is the perpendicular distance from the X-axis as shown below.

Thus we can easily find the distance of a point from the axes just by seeing the x and y coordinates. But, how to find the distance between two points when their coordinates are given? We use Pythagoras theorem to find the distance between two points.

Let P (x_{1},y_{1}) and Q (x_{2},y_{2}) be any two points in a plane. Let us find the distance between them PQ.

Then from the figure, we get

PR = |x_{2}-x_{1}| and QR = |y_{2}-y_{1}|.

We know that the Pythagoras theorem states, “In a right angled triangle, the square of the hypotenuse is the sum of the squares of the other two sides”.’

According to this theorem, we have

PQ^{2} = PR^{2} + QR^{2}

PQ^{2} = |x_{2}-x_{1}| ^{2} + |y_{2}-y_{1}|.^{2}

Taking square root on both sides, we get

PQ = √(|x_{2}-x_{1}| ^{2} + |y_{2}-y_{1}|^{2})

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