To find the distance between the points [tex]\((0, -1)\)[/tex] and [tex]\((3, -3)\)[/tex], we can use the distance formula, which is derived from the Pythagorean theorem. The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] in a Cartesian plane is given by:
[tex]\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Let's break down the steps to apply this formula to our specific points:
1. Identify the coordinates of the points:
- [tex]\((x_1, y_1) = (0, -1)\)[/tex]
- [tex]\((x_2, y_2) = (3, -3)\)[/tex]
2. Calculate the differences in the x-coordinates ([tex]\(dx\)[/tex]) and y-coordinates ([tex]\(dy\)[/tex]):
- [tex]\(dx = x_2 - x_1 = 3 - 0 = 3\)[/tex]
- [tex]\(dy = y_2 - y_1 = -3 - (-1) = -3 + 1 = -2\)[/tex]
Now we have the differences in the coordinates:
- [tex]\(dx = 3\)[/tex]
- [tex]\(dy = -2\)[/tex]
3. Plug these differences into the distance formula:
[tex]\[ d = \sqrt{(dx)^2 + (dy)^2} = \sqrt{3^2 + (-2)^2} \][/tex]
4. Calculate the squares of [tex]\(dx\)[/tex] and [tex]\(dy\)[/tex], then find their sum:
- [tex]\((dx)^2 = 3^2 = 9\)[/tex]
- [tex]\((dy)^2 = (-2)^2 = 4\)[/tex]
- Sum: [tex]\(9 + 4 = 13\)[/tex]
5. Finally, take the square root of the sum to find the distance:
[tex]\[ d = \sqrt{13} \][/tex]
So, the distance between the points [tex]\((0, -1)\)[/tex] and [tex]\((3, -3)\)[/tex] is [tex]\(\sqrt{13}\)[/tex].
Therefore, the correct answer is:
D. [tex]\(\sqrt{13}\)[/tex]
