Question

An image is drawn of a figure. After a 180 degree (counterclockwise) rotation about the origin,P(7,-2) is mapped to P'(-7,2). Identify the rule that maps P to P'

Answer 1

To determine the transformation rule that maps point \( P(7, -2) \) to point \( P'(-7, 2) \) after a 180-degree counterclockwise rotation about the origin, we will analyze the effect of such a rotation on the coordinates of points in the Cartesian plane.

The general rule for rotating a point \( (x, y) \) by 180 degrees counterclockwise around the origin is given by:

\[
(x, y) \rightarrow (-x, -y)
\]

Now, if we apply this rule to \( P(7, -2) \):

1. For the x-coordinate:
\[ -x = -7 \]
2. For the y-coordinate:
\[ -y = -(-2) = 2 \]

Thus, the rotation of \( P(7, -2) \) gives us:

\[
P'(-7, 2)
\]

This matches the point \( P' \) that you provided.

Therefore, the transformation rule that maps \( P(7, -2) \) to \( P'(-7, 2) \) via a 180-degree counterclockwise rotation about the origin is:

\[
(x, y) \rightarrow (-x, -y)
\]