Distance Formula

In this lesson, we’ll establish the formula to find out the distance between two points whose coordinates are given. This formula is commonly known as the Distance Formula.

Distance Formula

Consider two points P and Q, whose coordinates are given, say (x1, y1) and (x2, y2). We have to find the length of PQ, i.e. distance between the points P and Q.

Distance between two points

Fig. 1: Distance between two points

To compute the distance, we do some constructions as follows:

Distance Formula derivation

Fig. 2: Some constructions

QB & PA parallel to the Y axis, and QD & PC parallel to the X axis. CP is extended to meet QB at R.

Here’s the idea – if we’re able to find the lengths QR and PR, then we can apply Pythagoras theorem in triangle PQR to find PQ. That is PQ=\(\sqrt{PR^2 + QR^2}\)

Observe that QR = QB – RB and QB is the distance of Q from the X axis which is y2 (its ordinate)

PA is the distance of the point P from the X axis, which is (by definition) equal to its ordinate y1. And, PA = RB, as PR is parallel to the X axis.  Therefore RB = y1

Hence, QR = QB – RB = y– y1 and similarly, PR = x– x1

The following figure makes it clear

Distance Formula derivation

Fig. 3: The details

And.. we’re done ! Having obtained PR and QR, we can finally obtain the distance PQ as \( \sqrt{PR^2 + QR^2} \) or \(\sqrt{( x_2 – x_1)^2 + (y_2 – y_1)^2}\)


Lesson Summary

  1. The distance between two points whose coordinates are (x1, y1) and (x2, y2) is given by \( \sqrt{( x_2 – x_1)^2 + (y_2 – y_1)^2}\)

As an exercise, try to find out the coordinates of  R, A, B, C and D. The next lesson will cover some examples and applications.

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