Question

Show rotation 90 degrees counterclockwise

Answer 1

To rotate a point or shape 90 degrees counterclockwise around a specific point in a two-dimensional coordinate system, you can follow these steps:

1. **Identify the Point of Rotation**: Typically, this is the origin (0, 0), but it can be any point (h, k).

2. **Translate the Point (if necessary)**: If your rotation point is not the origin, translate the shape or point by subtracting the rotation point's coordinates.

3. **Apply the Rotation Formula**: For a point \((x, y)\), the new coordinates \((x', y')\) after a 90-degree counterclockwise rotation around the origin are given by:
\[
x' = -y
\]
\[
y' = x
\]

4. **Translate Back (if necessary)**: If you translated the points in step 2, add back the coordinates of the rotation point.

### Example
Let's say we want to rotate the point (3, 2) 90 degrees counterclockwise around the origin (0, 0).

1. **Original Point**: (3, 2)
2. **Rotation Formula**:
- New x-coordinate: \(-2\)
- New y-coordinate: \(3\)

So the rotated point is \((-2, 3)\).

### General Case
For a shape composed of multiple points:

1. Apply the rotation formula to each point in the shape.
2. Plot the new points to visualize the rotated shape.

If we had a triangle with vertices at (1, 2), (3, 4), and (5, 1), the rotated vertices would be:

- (1, 2) becomes (-2, 1)
- (3, 4) becomes (-4, 3)
- (5, 1) becomes (-1, 5)

Now the new vertices of the triangle after a 90-degree counterclockwise rotation around the origin are (-2, 1), (-4, 3), and (-1, 5).