# Distance between two points . What is the distance between the points (0, 5) and (3, 9) ?

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Since the problem provides the coordinates of two points, you need to use distance formula such that:

`d = sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2)`

Identifying `(x_1,y_1) = (0,5)` and `(x_2,y_2) = (3,9)` yields:

`d = sqrt((3 - 0)^2 + (9 - 5)^2)`

`d = sqrt(9 + 16) => d = sqrt 25 => d = 5`

**Hence, evaluating the distance between the given points, using distance formula, yields d = 5.**

The distance between any two points with coordinates (x, y) and (X, Y) is given by the formula `D = sqrt((X - x)^2 + (Y - y)^2)`

For the given points (0, 5) and (3, 9), X = 0, Y = 5, x = 3 and y = 9.

The distance between these points is:

D = `sqrt((0 - 3)^2 + (5 - 9)^2)`

= `sqrt(9 + 16)`

= `sqrt 25`

= 5

The distance between the points (0, 5) and (3, 9) 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.