To find the length of the segment [tex]\(\overline{ST}\)[/tex] with endpoints [tex]\(S(-7, -6)\)[/tex] and [tex]\(T(2, 4)\)[/tex], we'll use the distance formula, which is given by:
[tex]\[
\text{Distance} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}
\][/tex]
Here, the coordinates of point [tex]\(S\)[/tex] are [tex]\((-7, -6)\)[/tex] and the coordinates of point [tex]\(T\)[/tex] are [tex]\((2, 4)\)[/tex].
First, we calculate the differences in the coordinates:
[tex]\[
x_2 - x_1 = 2 - (-7) = 2 + 7 = 9
\][/tex]
[tex]\[
y_2 - y_1 = 4 - (-6) = 4 + 6 = 10
\][/tex]
Next, we square these differences:
[tex]\[
(9)^2 = 81
\][/tex]
[tex]\[
(10)^2 = 100
\][/tex]
Now, we add these squares:
[tex]\[
81 + 100 = 181
\][/tex]
Finally, we take the square root of this sum to get the distance:
[tex]\[
\sqrt{181} \approx 13.45362404707371
\][/tex]
So, the length of [tex]\(\overline{ST}\)[/tex] is:
[tex]\[
\sqrt{181}
\][/tex]
Thus, the correct answer is [tex]\(\boxed{\sqrt{181}}\)[/tex].
