In order to know the distance of two points we need to perform the Pythagorean Theorem which is:
[tex]a^2+b^2=c^2[/tex]
This is where a and b are the legs of the triangle and c is the hypotenuse. (This theorem only works in right triangles, triangles with an angle of measure 90°)
In a Cartesian plane the coordinates of two points would be [tex](x_a,y_a)[/tex] and [tex](x_b,y_b)[/tex] this would translate the Pythagorean Theorem into:
[tex](x_a-x_b)^2+(y_a-y_b)^2=c^2[/tex]
What we need in the problem is the value of c so we know that [tex](x_a,y_a)=(1,3) [/tex] and [tex](x_b,y_b)=(4,7)[/tex].
We substitute the values into the Pythagorean Theorem and get:
[tex]c^2=(1-4)^2+(3-7)^2 \\ =(-3)^3+(-4)^2 \\ =9+16 \\ =25[/tex]
We know now that the square of the distance of the points is 25 so the distance is either +5 or -5 but since a distance is always positive the distance is 5.
