What is the distance between the points (2, 3) and (5, 7)?

To find the distance between the points (2, 3) and (5, 7), we apply the distance formula, which is defined as:

[

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

]

In this scenario, the coordinates of the first point are ((x_1, y_1) = (2, 3)) and the second point are ((x_2, y_2) = (5, 7)).

Substituting these values into the formula gives us:

1. Calculate the change in the x-coordinates:

[

x_2 - x_1 = 5 - 2 = 3

]

2. Calculate the change in the y-coordinates:

[

y_2 - y_1 = 7 - 3 = 4

]

3. Now, substitute these results back into the distance formula:

[

d = \sqrt{(3)^2 + (4)^2}

]

4. Calculate the squares:

[

d = \sqrt{9 + 16}

]

5. Add the squared