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 follow these steps:
1. Identify the coordinates of points [tex]\(W\)[/tex] and [tex]\(X\)[/tex]:
- [tex]\(W = (5, -3)\)[/tex]
- [tex]\(X = (-1, -9)\)[/tex]
2. Calculate the differences in the x-coordinates and y-coordinates:
- [tex]\(\Delta x = X_x - W_x = -1 - 5 = -6\)[/tex]
- [tex]\(\Delta y = X_y - W_y = -9 - (-3) = -9 + 3 = -6\)[/tex]
3. Recall the distance formula for the length of a line segment between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex]:
[tex]\[
d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
4. Substitute the values [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the distance formula:
[tex]\[
d = \sqrt{(\Delta x)^2 + (\Delta y)^2} = \sqrt{(-6)^2 + (-6)^2}
\][/tex]
5. Simplify the expression:
[tex]\[
d = \sqrt{36 + 36} = \sqrt{72}
\][/tex]
6. Simplify [tex]\(\sqrt{72}\)[/tex]:
[tex]\[
\sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \times \sqrt{2} = 6\sqrt{2}
\][/tex]
7. Approximate the square root of 2 to obtain a numerical value:
[tex]\[
\sqrt{2} \approx 1.414
\][/tex]
Thus,
[tex]\[
6 \times 1.414 \approx 8.485
\][/tex]
So, the calculated length of [tex]\(\overline{WX}\)[/tex] is approximately [tex]\(8.485\)[/tex].
Therefore, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(\boxed{8.485}\)[/tex], not 6.
