To determine the length of the line segment [tex]\(\overline{WX}\)[/tex] with endpoints [tex]\(W(2, -7)\)[/tex] and [tex]\(X(5, -4)\)[/tex], we use the distance formula:
[tex]\[
\text{distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates are:
- [tex]\(W\)[/tex] with coordinates [tex]\((x_1, y_1) = (2, -7)\)[/tex]
- [tex]\(X\)[/tex] with coordinates [tex]\((x_2, y_2) = (5, -4)\)[/tex]
We can plug these values into the distance formula:
1. Compute the difference in the [tex]\(x\)[/tex]-coordinates:
[tex]\[
x_2 - x_1 = 5 - 2 = 3
\][/tex]
2. Compute the difference in the [tex]\(y\)[/tex]-coordinates:
[tex]\[
y_2 - y_1 = -4 - (-7) = -4 + 7 = 3
\][/tex]
3. Square these differences:
[tex]\[
(x_2 - x_1)^2 = 3^2 = 9
\][/tex]
[tex]\[
(y_2 - y_1)^2 = 3^2 = 9
\][/tex]
4. Sum these squared differences:
[tex]\[
(x_2 - x_1)^2 + (y_2 - y_1)^2 = 9 + 9 = 18
\][/tex]
5. Finally, take the square root of this sum to find the distance:
[tex]\[
\text{distance} = \sqrt{18} = \sqrt{9 \times 2} = 3\sqrt{2}
\][/tex]
Thus, the length of [tex]\(\overline{WX}\)[/tex] is [tex]\(3\sqrt{2}\)[/tex].
The correct answer is:
E. [tex]\(3\sqrt{2}\)[/tex]
