Whenever we need to find the distance of two point we are looking for the hypotenuse of a right triangle.
We need to use the Pythagorean Theorem to find the hypotenuse.
The Pythagorean Theorem is:
[tex]a^2+b^2=c^2[/tex]
This applies to only to right triangles. a and b are the side lengths of the legs while c is the length of the hypotenuse.
In a Cartesian plane the side lengths a and b are represented like this:
[tex](x_a-y_a)=a \\ (x_b-yb)=b[/tex]
So the Pythagorean Theorem would be:
[tex](x_a-y_a)^2+(x_b-y_b)^2=c^2[/tex]
We have [tex](x_a,y_a)[/tex] as [tex](0,2)[/tex]
and [tex](x_b,y_b)[/tex] as [tex](-2,0)[/tex]
We substitute the values to the Pythagorean theorem:
[tex]c^2=(0-(-2)^2+(2-0)^2 \\ =2^2+2^2 \\ =4+4 \\ =8[/tex]
[tex]c^2=8 \\ c= \sqrt{8} =2 \sqrt{2} [/tex]
The distance between the points cannot be negative so only the positive square root is considered. Therefore the distance between the points is [tex]2 \sqrt{2} [/tex] units.
