To determine the distance between the points [tex]\((-3, -1)\)[/tex] and [tex]\((-9, -10)\)[/tex], we can use the distance formula for points in a coordinate plane. The distance formula is derived from the Pythagorean Theorem and is given as:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here [tex]\((x_1, y_1)\)[/tex] are the coordinates of the first point and [tex]\((x_2, y_2)\)[/tex] are the coordinates of the second point.
Let's plug in the given coordinates:
- [tex]\((x_1, y_1) = (-3, -1)\)[/tex]
- [tex]\((x_2, y_2) = (-9, -10)\)[/tex]
First, find the differences in the [tex]\(x\)[/tex]-coordinates and [tex]\(y\)[/tex]-coordinates:
[tex]\[
x_2 - x_1 = -9 - (-3) = -9 + 3 = -6
\][/tex]
[tex]\[
y_2 - y_1 = -10 - (-1) = -10 + 1 = -9
\][/tex]
Next, square these differences:
[tex]\[
(-6)^2 = 36
\][/tex]
[tex]\[
(-9)^2 = 81
\][/tex]
Now, sum these squared differences:
[tex]\[
36 + 81 = 117
\][/tex]
Finally, take the square root of this sum to find the distance:
[tex]\[
d = \sqrt{117}
\][/tex]
So, the distance between the points [tex]\((-3, -1)\)[/tex] and [tex]\((-9, -10)\)[/tex] is [tex]\(\sqrt{117}\)[/tex] units.
Therefore, the correct answer is:
[tex]\[
\boxed{\sqrt{117}}
\][/tex]
