Let's determine the coordinates of the vertices of [tex]\( \Delta G'H'I' \)[/tex] after a [tex]\( 270^\circ \)[/tex] counterclockwise rotation about the origin.
1. Understand the effect of a [tex]\( 270^\circ \)[/tex] counterclockwise rotation:
When a point [tex]\((x, y)\)[/tex] is rotated [tex]\( 270^\circ \)[/tex] counterclockwise around the origin, the new coordinates [tex]\((x', y')\)[/tex] can be determined by:
[tex]\[
(x', y') = (y, -x)
\][/tex]
2. Apply the rotation to each vertex of [tex]\( \Delta GHI \)[/tex]:
- For [tex]\( G(-4, 0) \)[/tex]:
[tex]\[
(x', y') = (0, -(-4)) = (0, 4)
\][/tex]
So, [tex]\( G' = (0, 4) \)[/tex].
- For [tex]\( H(5, 2) \)[/tex]:
[tex]\[
(x', y') = (2, -5) = (2, -5)
\][/tex]
So, [tex]\( H' = (2, -5) \)[/tex].
- For [tex]\( I(-2, -3) \)[/tex]:
[tex]\[
(x', y') = (-3, -(-2)) = (-3, 2)
\][/tex]
So, [tex]\( I' = (-3, 2) \)[/tex].
3. Summary of the coordinates after rotation:
The coordinates of the vertices of [tex]\( \Delta G'H'I' \)[/tex] after a [tex]\( 270^\circ \)[/tex] counterclockwise rotation about the origin are:
[tex]\[
G' = (0, 4),\ H' = (2, -5),\ I' = (-3, 2)
\][/tex]
So, the vertices of [tex]\( \Delta G'H'I' \)[/tex] are [tex]\( (0, 4), (2, -5), \)[/tex] and [tex]\( (-3, 2) \)[/tex].
