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 (x_{1}, y_{1}) and (x_{2}, y_{2}). We have to find the length of PQ, i.e. distance between the points P and Q.

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

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 y_{2} (its ordinate)

PA is the distance of the point P from the X axis, which is (by definition) equal to its ordinate y_{1}. And, PA = RB, as PR is parallel to the X axis. Therefore RB = y_{1}

Hence, QR = QB – RB = y_{2 }– y_{1} and similarly, PR = x_{2 }– x_{1}

The following figure makes it clear

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

- The distance between two points whose coordinates are (x
_{1}, y_{1}) and (x_{2}, y_{2}) 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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