Distance Between Points

Video lesson · from 0:281 worked example

One lesson video covers all of The Straight Line, so it opens at 0:28 for this topic — not from the beginning.

To calculate the distance between two points, we can construct a right-angled triangle and use Pythagoras' Theorem.

The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

Leave the answer as a surd: the distance formula usually gives a root. Simplify it rather than rounding, especially when the result feeds into later working.
Squaring kills the sign: so the order of the points does not matter — but brackets around a negative difference do.
It is Pythagoras: if you forget the formula, sketch the right-angled triangle and use $a^2+b^2=c^2$ .
In three dimensions add the third term: the pattern extends to $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}$ .

Example 1

Plot the coordinates $(1,3)$ and $(6,9)$ . Using Pythagoras' Theorem, calculate the distance between these two points.

Label the points: $(x_1 = 1, y_1 = 3)$ and $(x_2 = 6, y_2 = 9)$ .

d^2 = (x_2 - x_1)^2 + (y_2 - y_1)^2

d^2 = (6 - 1)^2 + (9 - 3)^2

d^2 = 5^2 + 6^2

d^2 = 25 + 36 = 61

d = \sqrt{61}

The distance is $\sqrt{61}$ .