To find the distance between the two points [tex]\((3, -7)\)[/tex] and [tex]\((-2, -2)\)[/tex], we can use the distance formula, which is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Let's break it down step by step:
1. Identify the coordinates of the points:
- Point 1: [tex]\((x_1, y_1) = (3, -7)\)[/tex]
- Point 2: [tex]\((x_2, y_2) = (-2, -2)\)[/tex]
2. Calculate the differences in the coordinates:
- [tex]\(\Delta x = x_2 - x_1 = -2 - 3 = -5\)[/tex]
- [tex]\(\Delta y = y_2 - y_1 = -2 - (-7) = -2 + 7 = 5\)[/tex]
3. Substitute the values into the distance formula:
[tex]\[
d = \sqrt{(-5)^2 + (5)^2}
\][/tex]
4. Calculate the squares of the differences:
- [tex]\((-5)^2 = 25\)[/tex]
- [tex]\(5^2 = 25\)[/tex]
5. Add the squares:
[tex]\[
(-5)^2 + 5^2 = 25 + 25 = 50
\][/tex]
6. Take the square root of the sum:
[tex]\[
d = \sqrt{50}
\][/tex]
7. Simplify the square root (finding simplest radical form):
We can simplify [tex]\(\sqrt{50}\)[/tex] by expressing it as a product of square roots of its factors:
[tex]\[
\sqrt{50} = \sqrt{25 \cdot 2} = \sqrt{25} \cdot \sqrt{2} = 5\sqrt{2}
\][/tex]
Therefore, the distance between the points [tex]\((3, -7)\)[/tex] and [tex]\((-2, -2)\)[/tex] in simplest radical form is:
[tex]\[
\boxed{5\sqrt{2}}
\][/tex]
