To determine the length of the line segment [tex]\(\overline{WX}\)[/tex] with endpoints [tex]\(W(-5, -1)\)[/tex] and [tex]\(X(2, 6)\)[/tex], we use the distance formula. The distance formula for the length between 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]
Let's apply the coordinates of [tex]\(W\)[/tex] and [tex]\(X\)[/tex] into this formula.
1. Calculate the differences in the x-coordinates and y-coordinates:
[tex]\[
\Delta x = x_2 - x_1 = 2 - (-5) = 2 + 5 = 7
\][/tex]
[tex]\[
\Delta y = y_2 - y_1 = 6 - (-1) = 6 + 1 = 7
\][/tex]
2. Substitute the differences [tex]\(\Delta x\)[/tex] and [tex]\(\Delta y\)[/tex] into the distance formula:
[tex]\[
\text{Distance} = \sqrt{(7)^2 + (7)^2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{49 + 49}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{98}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{49 \times 2}
\][/tex]
[tex]\[
\text{Distance} = \sqrt{49} \times \sqrt{2}
\][/tex]
[tex]\[
\text{Distance} = 7 \sqrt{2}
\][/tex]
Therefore, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(7 \sqrt{2}\)[/tex].
The correct answer is:
D. [tex]\(7 \sqrt{2}\)[/tex]
