Question

How do you find the distance between `(2,5)` and `(-1,-5)`?

Answer 1

Using the Pythagorean Theorem.

Explanation:

The distance between two points is the segment between the two points.
This segment is the hypotenuse of a right triangle, so its length has to be calculated using the Pythagorean Theorem.
The other two sides of this triangle are given by the difference of the coordinates of the points.

So the first side is given by `2-(-1)=2+1=3`
The second side is `5-(-5)=10`.
For the final result it is not important the order of the points, we can calculate in the other order. For example we can say that the first side is `-1-2=-3` and the second side is `-5-5=-10`.
We have a negative number, but in the Pythagorean Theorem we calculate the square of these numbers, so the final result is still correct.

The distance is then `d=sqrt(3^2+10^2)` or, using the other difference `d=sqrt((-3)^2+(-10)^2)`. The result is the same

`d=sqrt(9+100)=sqrt(109\approx)10.44`.

We can write the general rule for the distance of two points `p_1` of coordinates `(x_1, y_1)` and `p_2` of coordinates `(x_2, y_2)` as

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