To find the length of the line segment [tex]\(\overline{W X}\)[/tex] where [tex]\(W\)[/tex] has coordinates [tex]\((5, -3)\)[/tex] and [tex]\(X\)[/tex] has coordinates [tex]\((-1, -9)\)[/tex], we can use the distance formula between two points in a coordinate plane. The distance formula is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
In our case, let [tex]\((x_1, y_1) = (5, -3)\)[/tex] and [tex]\((x_2, y_2) = (-1, -9)\)[/tex]. Plugging in these values, we get:
[tex]\[
\text{Distance} = \sqrt{((-1) - 5)^2 + ((-9) - (-3))^2}
\][/tex]
Simplifying the expressions within the parentheses:
[tex]\[
\text{Distance} = \sqrt{(-1 - 5)^2 + (-9 + 3)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{(-6)^2 + (-6)^2}
\][/tex]
Squaring the numbers inside the square root:
[tex]\[
\text{Distance} = \sqrt{36 + 36}
\][/tex]
Combining the results:
[tex]\[
\text{Distance} = \sqrt{72}
\][/tex]
We know that [tex]\(72\)[/tex] can be written as [tex]\(36 \times 2\)[/tex]:
[tex]\[
\text{Distance} = \sqrt{36 \times 2}
\][/tex]
This can be further simplified:
[tex]\[
\text{Distance} = \sqrt{36} \times \sqrt{2}
\][/tex]
[tex]\[
\text{Distance} = 6 \sqrt{2}
\][/tex]
So, the length of [tex]\(\overline{W X}\)[/tex] is:
[tex]\[
\boxed{6 \sqrt{2}}
\][/tex]
