To find the distance of two points we need to use the Pythagorean Theorem the distance between points is a hypotenuse of a right triangle.
The Pythagorean Theorem states that:
[tex]a^2+b^2=c^2[/tex]
The Pythagorean Theorem triangles with 90° (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 the coordinates of point A which is [tex](-2,3)[/tex]
and [tex](x_b,y_b)[/tex] as the coordinates of point B which is [tex](4,1)[/tex]
We substitute the values to the Pythagorean theorem:
[tex]c^2=(-2-4)^2+(3-1)^2 \\ =(-6)^2+(2)^2 \\ =36+4 \\ =40[/tex]
[tex]c= \sqrt{40} =2 \sqrt{10} [/tex]
Therefore the length of the line segment is [tex]2 \sqrt{10} [/tex]
