To find the length of the line segment [tex]\(\overline{WX}\)[/tex] with endpoints [tex]\(W(5, -3)\)[/tex] and [tex]\(X(-1, -9)\)[/tex], we can use the distance formula. The distance formula 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]
Substitute the coordinates of the points [tex]\(W(5, -3)\)[/tex] and [tex]\(X(-1, -9)\)[/tex] into the formula:
[tex]\[
d = \sqrt{((-1) - 5)^2 + ((-9) - (-3))^2}
\][/tex]
Calculate the differences in the coordinates:
[tex]\[
x_2 - x_1 = -1 - 5 = -6
\][/tex]
[tex]\[
y_2 - y_1 = -9 - (-3) = -9 + 3 = -6
\][/tex]
Now substitute these differences into the distance formula:
[tex]\[
d = \sqrt{(-6)^2 + (-6)^2}
\][/tex]
Calculate the squares of [tex]\(-6\)[/tex]:
[tex]\[
(-6)^2 = 36
\][/tex]
[tex]\[
(-6)^2 = 36
\][/tex]
Sum the squares:
[tex]\[
36 + 36 = 72
\][/tex]
Finally, take the square root of 72:
[tex]\[
d = \sqrt{72} = \sqrt{36 \times 2} = 6\sqrt{2}
\][/tex]
Thus, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(6\sqrt{2}\)[/tex], which corresponds to option E. Therefore, the correct answer is:
E. [tex]\(6\sqrt{2}\)[/tex]
