The image shows a diagram where the quadrilateral ABCD is rotated 90° counterclockwise about the origin, denoted as $r(90^\circ, O)(ABCD)$. The task is to determine the coordinates of points A, B, C, and D after the rotation.

Step-by-step solution:

When a point $(x, y)$ is rotated by 90° counterclockwise about the origin, the new coordinates become $(-y, x)$.

Let's apply this to the given points from the diagram:

1. Original coordinates:

• Point A: $(4, 2)$
• Point B: $(3, -1)$
• Point C: $(0, -2)$
• Point D: $(1, 3)$

2. After 90° counterclockwise rotation:

• For Point A $(4, 2)$, the new coordinates are $(-2, 4)$.
• For Point B $(3, -1)$, the new coordinates are $(1, 3)$.
• For Point C $(0, -2)$, the new coordinates are $(2, 0)$.
• For Point D $(1, 3)$, the new coordinates are $(-3, 1)$.

Final coordinates after the rotation:

• Point A: $(-2, 4)$
• Point B: $(1, 3)$
• Point C: $(2, 0)$
• Point D: $(-3, 1)$

Would you like further details or have any questions?

5 Related Questions:

1. How would the coordinates change for a 180° rotation around the origin?
2. What is the general formula for a 270° counterclockwise rotation?
3. How do we determine the center of rotation if not given as the origin?
4. Can we use matrices to represent 2D rotations? How?
5. How would a clockwise rotation differ in terms of coordinate transformations?

Tip:

When performing 2D rotations, visualizing the change by drawing or tracing a small portion of the graph can help clarify the transformation effect.