To determine the coordinates of the image vertices [tex]\( B' \)[/tex] when the preimage vertex [tex]\( B \)[/tex] is rotated 180 degrees counterclockwise, we can follow these steps:
1. Identify the coordinates of the preimage point [tex]\( B \)[/tex]: The given coordinates for point [tex]\( B \)[/tex] are [tex]\( B(1, 0) \)[/tex].
2. Know the rotation rule: When a point [tex]\((x, y)\)[/tex] is rotated 180 degrees counterclockwise around the origin, the new coordinates [tex]\((x', y')\)[/tex] are given by:
[tex]\[
(x, y) \rightarrow (-x, -y)
\][/tex]
3. Apply the rotation rule to the coordinates of point [tex]\( B \)[/tex]:
- The original coordinates of [tex]\( B \)[/tex] are [tex]\( B(1, 0) \)[/tex].
- Applying the 180-degree rotation transformation:
[tex]\[
(1, 0) \rightarrow (-1, -0)
\][/tex]
4. Simplify the result: The coordinate [tex]\( -0 \)[/tex] is equivalent to [tex]\( 0 \)[/tex]. So the new coordinates of [tex]\( B' \)[/tex] after 180-degree rotation are:
[tex]\[
B'( -1, 0)
\][/tex]
Therefore, the image vertex of [tex]\( B' \)[/tex] after a 180-degree counterclockwise rotation is [tex]\( \boxed{(-1, 0)} \)[/tex].
