# 1.5 Distance between two points in the plane

Next, we find the formula for the distance between two points *P* (*x*_{1}, *y*_{1}) and *Q*(*x*_{2}, *y*_{2}) in the plane. In the diagram below we have drawn *P* and *Q* in the first quadrant, but the formula we derive holds wherever the points are in the plane.

We can construct a right-angled triangle *PNQ* as shown; the line *PN* is parallel to the *x*-axis, the line *QN* is parallel to the *y*-axis, the angle *PNQ* is a right angle, and *PQ* is the hypotenuse of the triangle.

The length of *PN* is |*x*_{2} − *x*_{1}| and the length of *QN* is |*y*_{2} − *y*_{1}|.

*Note:* For any real number *x*,

It follows from Pythagoras' Theorem that *PQ*^{2} = *PN*^{2} + *QN*^{2}, so

## Distance Formula in two-dimensional Euclidean space

The distance between two points (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) in the plane is

For example, it follows from the formula above that the distance between the points (1, 2) and (3, −4) is

## Example 5

Find the distance between each of the following pairs of points in the plane.

**(a)**(0, 0) and (5, 0).**(b)**(0, 0) and (3, 4).**(c)**(1, 2) and (5, 1).**(d)**(3, −8) and (−1, 4).

### Answer

We use the formula for the distance between two points in the plane. This gives the following distances.

**(a)****(b)****(c)****(d)**