https://file.aiquickdraw.com/mathbot/file/17302031011640ic7hdlp.jpg

To solve this problem, we need to perform the following sequence of transformations on point \( R \):

1. **Rotate point \( R \) by 90° about the origin**.
2. **Reflect the resulting point across the line \( x = 0 \) (the y-axis)**.

Let's work through each step using point \( R \)'s coordinates.

### Step 1: Initial Coordinates of \( R \)
From the graph, the coordinates of point \( R \) are:
\[
R = (3, -1)
\]

### Step 2: Rotation 90° About the Origin
A 90° rotation about the origin transforms a point \( (x, y) \) to \( (-y, x) \).

Applying this to \( R = (3, -1) \):
\[
R' = (-(-1), 3) = (1, 3)
\]

So, after a 90° rotation, the coordinates of \( R' \) are:
\[
R' = (1, 3)
\]

### Step 3: Reflection Across \( x = 0 \) (the y-axis)
Reflecting a point across the y-axis changes the sign of the x-coordinate. For a point \( (x, y) \), the reflection across \( x = 0 \) becomes \( (-x, y) \).

Applying this to \( R' = (1, 3) \):
\[
R'' = (-1, 3)
\]

### Final Answer
After the sequence of transformations, the coordinates of \( R' \) are:
\[
R' = (-1, 3)
\]

Would you like further details, or do you have any questions about this solution?

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Here are some follow-up questions related to transformations and rotations:

1. What are the coordinates of point \( Q \) after the same transformations?
2. How would the coordinates change if the rotation was 180° instead of 90°?
3. How do the transformations differ if we reflect across the line \( y = 0 \) instead?
4. What sequence of transformations would map point \( T \) to \( R \)?
5. How can we represent these transformations as matrix operations?

**Tip:** When rotating points about the origin, remember that a 90° rotation swaps the coordinates and changes one sign based on the direction of rotation.

For items 7-10, use the figure shown. Find the coordinates of the specified vertex after the given sequence of transformations: a rotation 90° about the origin, then a reflection across x = 0.

Step-by-step solution to find the new coordinates of vertex R after a 90° rotation about the origin followed by a reflection across the y-axis. This problem explores coordinate geometry concepts and transformation formulas, suitable for Grades 8-10.

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