To determine the coordinates of point [tex]\( M' \)[/tex] after rotating point [tex]\( M(5, 6) \)[/tex] by [tex]\( 90^{\circ} \)[/tex] counterclockwise, we can use the standard rotation formula for [tex]\( 90^{\circ} \)[/tex] counterclockwise rotation.
When a point [tex]\((x, y)\)[/tex] is rotated [tex]\( 90^{\circ} \)[/tex] counterclockwise around the origin, the new coordinates [tex]\((x', y')\)[/tex] can be found using the following transformation:
[tex]\[ x' = -y \][/tex]
[tex]\[ y' = x \][/tex]
Let's apply this transformation to the point [tex]\( M(5, 6) \)[/tex]:
1. Start with the original coordinates [tex]\( (5, 6) \)[/tex].
2. Apply the transformation:
[tex]\[ x' = -y = -6 \][/tex]
[tex]\[ y' = x = 5 \][/tex]
Therefore, the new coordinates of the rotated point [tex]\( M' \)[/tex] are [tex]\((-6, 5)\)[/tex].
So, the ordered pair for [tex]\( M' \)[/tex] after the [tex]\( 90^{\circ} \)[/tex] counterclockwise rotation is:
[tex]\[ \boxed{(-6, 5)} \][/tex]
Out of the given options:
- [tex]\( M'(-6, 5) \)[/tex]
- [tex]\( M'(6, -5) \)[/tex]
- [tex]\( M'(5, -6) \)[/tex]
- [tex]\( M'(-5, 6) \)[/tex]
The correct answer is [tex]\( M'(-6, 5) \)[/tex].
