To find the pre-image of a point P' after a 270° rotation about the origin, we need to determine how a point transforms under that specific rotation.
A 270° rotation counter-clockwise is equivalent to a 90° rotation clockwise. The transformation for a point \((x, y)\) under a 90° clockwise rotation about the origin is given by the following rule:
\[ (x, y) \rightarrow (y, -x) \]
Given the image point P' is \((-2, 7)\), we denote this as \((x', y')\):
- \(x' = -2\)
- \(y' = 7\)
Now, we will apply the inverse transformation (90° counter-clockwise) to find the pre-image:
To find the pre-image, we use the transformation for a 90° counter-clockwise rotation, which is: \[ (x, y) \rightarrow (-y, x) \]
Thus, for the image point P' \((-2, 7)\):
- Substitute \(y' = 7\):
- \(-y' = -7\)
- Substitute \(x' = -2\):
- \(x' = -2\)
Now, we find the pre-image: \[ (-2, 7) \rightarrow (-7, -2) \]
So, the pre-image point P before the 270° rotation about the origin is \((-7, -2)\).