Question

The image of a point on a coordinate plane is at P' (-2,7). Find the pre-image before a 270° rotation about the origin

Answer 1

Wouldn't it be (7,2)?

Answer 2

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' \((-2, 7)\):

1. Substitute \(y' = 7\):
- \(-y' = -7\)
2. 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 \((-7, -2)\).