To determine the signs of the coordinates of point $P'$, we need to follow the transformation described:

Step 1: Understand the 90-degree counterclockwise rotation.

When a point $(x, y)$ is rotated 90 degrees counterclockwise around the origin, its new coordinates $(x', y')$ are given by: $(x', y') = (-y, x)$ This means:

• The x-coordinate of the new point $P'$ will be the negative of the y-coordinate of the original point $P$.
• The y-coordinate of the new point $P'$ will be the x-coordinate of the original point $P$.

Step 2: Analyze the signs of the coordinates after rotation.

Let's consider each quadrant and how the signs change:

• If the original point $P$ is in Quadrant I (where both coordinates are positive), after rotation, $P'$ will be in Quadrant II, where the x-coordinate is negative and the y-coordinate is positive.
• If $P$ is in Quadrant II (x is negative, y is positive), after rotation, $P'$ will be in Quadrant III (both x and y are negative).
• If $P$ is in Quadrant III (both x and y are negative), after rotation, $P'$ will be in Quadrant IV (x is positive, y is negative).
• If $P$ is in Quadrant IV (x is positive, y is negative), after rotation, $P'$ will be in Quadrant I (both x and y are positive).

Since we don’t have the exact coordinates of $P$, the signs of $P'$'s coordinates depend on which quadrant $P$ is originally located in. But we can conclude from the general rule that a 90-degree counterclockwise rotation changes the signs in this way:

• If $P$ is in Quadrant I, then after the rotation, x is negative and y is positive.

Thus, the answer is B: x is negative, y is positive.

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Do you need more details or have any questions?

Here are 5 related questions you might consider:

1. How does rotating a point by 180 degrees affect its coordinates?
2. What are the coordinates of a point after a 270-degree counterclockwise rotation?
3. How does reflection across the x-axis change a point's coordinates?
4. What are the coordinates of a point after a reflection across the y-axis?
5. How would you find the coordinates of $P''$ after both a 90-degree rotation and reflection?

Tip: Always keep track of the quadrant your point moves to when applying transformations like rotations or reflections to easily determine sign changes.