**Triangle PQR is rotated 90 degrees counterclockwise around the origin to form P'Q'R'. The triangle P'Q'R' is reflected across the x-axis to form P''Q''R''.**

1

(a)

**What are the signs of the coordinates of (x, y) of point P'? (Rotate point P 90 degrees counterclockwise to find your answer.)**

A

Both x and y are positive 

B

x is negative, y is positive

C

x is positive, y is negative

D

Both x and y are negative

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.

Triangle PQR is rotated 90 degrees counterclockwise around the origin to form P'Q'R'. The triangle P'Q'R' is reflected across the x-axis to form P''Q''R''. What are the signs of the coordinates of (x, y) of point P'? (Rotate point P 90 degrees counterclockwise to find your answer.)

Learn how to determine the signs of the coordinates of a point after a 90-degree counterclockwise rotation around the origin. This problem involves geometric transformations and is suitable for students in Grades 8-10. Explore the effects of rotation and reflection on a triangle's coordinates.

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