The problem provides the information that you need to use distance formula to evaluate the distance between the given points, such that:
`d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`
Identifying `(x_1,y_1) = (5,2)` and `(x_2,y_2) = (0,-4)` yields:
`d = sqrt((0 - 5)^2 + (- 4 - 2)^2)`
`d = sqrt(25 + 36) => d = sqrt 61`
Hence, evaluating the distance between the given points, using distance formula, yields `d = sqrt 61.`
To determine the distance between 2 given points in the rectangular plane, we'll apply the Pythagorean theorem in the right angle triangle formed by the projections of the given points.
We'll note the points as A(5, 2) and B(0,-4).
The right angle triangle is ACB, where <C = 90 degrees and AB is the hypothenuse.
We'll calculate the cathetus AC:
AC = xA - xC
AC = 5 - 0
AC = 5
BC = yB - yC = 4 + 2 = 6
The hypothenuse AB:
AB^2 = AC^2 + BC^2
AB^2 = 5^2 + 6^2
AB^2 = 25 + 36
AB^2 = 61
AB = sqrt 61 units
We'll keep just the positive value, since AB represents a distance.