To determine the length of the line segment [tex]\(\overline{WX}\)[/tex] with endpoints [tex]\(W(5, -3)\)[/tex] and [tex]\(X(-1, -9)\)[/tex], we use the distance formula. The distance formula for two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is given by:
[tex]\[ \text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \][/tex]
Here, [tex]\(W = (5, -3)\)[/tex] and [tex]\(X = (-1, -9)\)[/tex].
1. Identify the coordinates:
[tex]\[
x_1 = 5, \quad y_1 = -3, \quad x_2 = -1, \quad y_2 = -9
\][/tex]
2. Substitute the coordinates into the distance formula:
[tex]\[
\sqrt{(-1 - 5)^2 + (-9 - (-3))^2}
\][/tex]
3. Simplify the expressions within the parentheses:
[tex]\[
-1 - 5 = -6 \quad \text{and} \quad -9 - (-3) = -9 + 3 = -6
\][/tex]
4. Substitute these values back into the formula:
[tex]\[
\sqrt{(-6)^2 + (-6)^2}
\][/tex]
5. Compute the squares:
[tex]\[
(-6)^2 = 36 \quad \text{and} \quad (-6)^2 = 36
\][/tex]
6. Add the results:
[tex]\[
36 + 36 = 72
\][/tex]
7. Finally, take the square root of the sum:
[tex]\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}
\][/tex]
Thus, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(\boxed{6\sqrt{2}}\)[/tex], which corresponds to option E.
