To determine the vertices of the image [tex]\( L^{\prime}M^{\prime}N^{\prime} \)[/tex] after rotating the preimage [tex]\( LMN \)[/tex] 90 degrees clockwise about the origin, we follow these steps for each vertex:
1. Start with the coordinates of each vertex of the triangle [tex]\( LMN \)[/tex]:
- [tex]\( L(-1, 5) \)[/tex]
- [tex]\( M(-1, 0) \)[/tex]
- [tex]\( N(-2, 5) \)[/tex]
2. Apply the transformation rule for a 90-degree clockwise rotation about the origin. This rule changes a point [tex]\((x, y) \)[/tex] to [tex]\((y, -x) \)[/tex].
Using this transformation rule:
- For vertex [tex]\( L(-1, 5) \)[/tex]:
[tex]\[
L^{\prime} = (y, -x) = (5, -(-1)) = (5, 1)
\][/tex]
- For vertex [tex]\( M(-1, 0) \)[/tex]:
[tex]\[
M^{\prime} = (y, -x) = (0, -(-1)) = (0, 1)
\][/tex]
- For vertex [tex]\( N(-2, 5) \)[/tex]:
[tex]\[
N^{\prime} = (y, -x) = (5, -(-2)) = (5, 2)
\][/tex]
3. Compile the new coordinates of the vertices:
- [tex]\( L^{\prime}(5, 1) \)[/tex]
- [tex]\( M^{\prime}(0, 1) \)[/tex]
- [tex]\( N^{\prime}(5, 2) \)[/tex]
Thus, the vertices of the image [tex]\( L^{\prime}M^{\prime}N^{\prime} \)[/tex], after a 90-degree clockwise rotation, are:
[tex]\[
L^{\prime}(5, 1), M^{\prime}(0, 1), N^{\prime}(5, 2)
\][/tex]
Therefore, the correct option is:
- [tex]\( L^{\prime}(5,1), M^{\prime}(0,1), N^{\prime}(5,2) \)[/tex]
