To find the length of the line segment \( \overline{BD} \) in the polygon defined by the given coordinates, we first need to identify the points \( B \) and \( D \). From the given vertices of the polygon:
- \( A(-7, 7) \)
- \( B(-3, 6) \)
- \( C(-3, 1) \)
- \( D(-7, 1) \)
We can see that point \( B \) corresponds to \( (-3, 6) \) and point \( D \) corresponds to \( (-7, 1) \).
Next, we can use the distance formula to find the length of the segment \( \overline{BD} \):
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
Substituting the coordinates of \( B \) and \( D \):
- \( B(x_1, y_1) = (-3, 6) \)
- \( D(x_2, y_2) = (-7, 1) \)
So we have:
\[ d = \sqrt{((-7) - (-3))^2 + (1 - 6)^2} \] \[ = \sqrt{(-7 + 3)^2 + (1 - 6)^2} \] \[ = \sqrt{(-4)^2 + (-5)^2} \] \[ = \sqrt{16 + 25} \] \[ = \sqrt{41} \]
The approximate value of \( \sqrt{41} \) is about \( 6.4 \), but since none of the options specifically match this result, it seems there might have been an error in assumptions or options.
However, based on the calculated distance of \( \overline{BD} \), we can conclude:
None of the provided options seem to match.