The problem asks which of the given rotations about the origin will transform the point $A(-X, y)$ to the point $A'(X, -y)$.

1. Problem Description

The problem asks which of the given rotations about the origin will transform the point $A(-X, y)$ to the point $A'(X, -y)$.

2. Solution Steps

We need to determine the rotation that maps $(-X, y)$ to $(X, -y)$.

Let's analyze each option:

(a) $R(O, -90^\circ)$: A rotation of $-90^\circ$ (or $270^\circ$) about the origin transforms a point $(x, y)$ to $(y, -x)$. Applying this to $(-X, y)$, we get $(y, -(-X)) = (y, X)$, which is not $(X, -y)$.

(b) $R(O, 90^\circ)$: A rotation of $90^\circ$ about the origin transforms a point $(x, y)$ to $(-y, x)$. Applying this to $(-X, y)$, we get $(-y, -X)$, which is not $(X, -y)$.

(c) $R(O, 180^\circ)$: A rotation of $180^\circ$ about the origin transforms a point $(x, y)$ to $(-x, -y)$. Applying this to $(-X, y)$, we get $(-(-X), -y) = (X, -y)$, which is what we want.

(d) $R(O, 360^\circ)$: A rotation of $360^\circ$ about the origin transforms a point $(x, y)$ to $(x, y)$. Applying this to $(-X, y)$, we get $(-X, y)$, which is not $(X, -y)$.

Therefore, the rotation that transforms $(-X, y)$ to $(X, -y)$ is a rotation of $180^\circ$ about the origin.

3. Final Answer

(c) R (O, 180°)