To find the distance between the points [tex]\((-6, 2)\)[/tex] and [tex]\( (0, -1) \)[/tex] using the distance formula, follow these steps:
1. Identify the coordinates:
Let [tex]\((x_1, y_1)\)[/tex] be [tex]\((-6, 2)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] be [tex]\((0, -1)\)[/tex].
2. Write down the distance formula:
The distance [tex]\(d\)[/tex] between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
3. Substitute the coordinates into the formula:
Substitute [tex]\(x_1 = -6\)[/tex], [tex]\(y_1 = 2\)[/tex], [tex]\(x_2 = 0\)[/tex], and [tex]\(y_2 = -1\)[/tex] into the formula:
[tex]\[
d = \sqrt{(0 - (-6))^2 + (-1 - 2)^2}
\][/tex]
Simplify the expressions inside the parentheses:
[tex]\[
d = \sqrt{(0 + 6)^2 + (-1 - 2)^2}
\][/tex]
[tex]\[
d = \sqrt{6^2 + (-3)^2}
\][/tex]
4. Calculate the squares inside the square root:
[tex]\[
d = \sqrt{36 + 9}
\][/tex]
5. Add the results:
[tex]\[
d = \sqrt{45}
\][/tex]
6. Simplify the square root if possible:
The number [tex]\(45\)[/tex] can be factored as [tex]\(45 = 9 \cdot 5\)[/tex]. Since [tex]\(9\)[/tex] is a perfect square ([tex]\(9 = 3^2\)[/tex]):
[tex]\[
d = \sqrt{9 \cdot 5} = \sqrt{9} \cdot \sqrt{5} = 3\sqrt{5}
\][/tex]
Therefore, the distance between the points [tex]\((-6, 2)\)[/tex] and [tex]\((0, -1)\)[/tex] is [tex]\(3\sqrt{5}\)[/tex].
