Let the points be A= ( 0,5) and B = (3,9)

To find the distance between the points A and B, we will use the distance formula as follows:

The distance Between A and B is given by:

D ( AB) = sqrt[( xB-xA)^2 + ( yB-yA)^2].

Let us substitute with the coordinates of A and B.

==> D ( AB) = sqrt[ ( 3-0)^2 + ( 9-5)^2]

==> D(AB) = sqrt( 3^2 + 4^2)

==> D(AB) = sqrt(9+ 16).

==> D(AB) = sqrt25.

==> D(AB) = 5

**Then, the distance between A and B is 5 units.**

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(0,5) and B(3,9).

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 = 0 - 3

AC = -3

BC = yB - yC

BC = 9 - 5

BC = 4

The hypothenuse AB:

AB^2 = AC^2 + BC^2

AB^2 = (-3)^2 + 4^2

AB^2 = 9 + 16

AB^2 = 25

**AB = sqrt 25**

**AB = 5**** units**

**We'll keep just the positive value, since AB represents a distance.**

The distance d between the two ponts (x1,y1) and (x2,y2) is given by:

d = sqrt{(x2-x1)^2+(y2-y1)^2}.

So the distance between (0,5) and (3,9) is given by:

d = {(3-0)^2 +(9 - 5)^2} = sqrt{3 + 4^2}.

d = sqrt{9+16} = sqrt25

d = 5.

Therefore the distance between (0,5) and (3,9) is 5.