To solve the problem of finding the new coordinates after a 180° counterclockwise rotation around the origin, we need to apply a specific transformation to each vertex of the shape. When a point $(x, y)$ is rotated 180° around the origin, its new coordinates become $(-x, -y)$.

Let's identify the coordinates of the vertices $B, C, D,$ and $E$ from the given image and calculate their new positions:

Original Coordinates:

• B: $(-2, 4)$
• C: $(2, 6)$
• D: $(2, 8)$
• E: $(-2, 6)$

New Coordinates after 180° Rotation:

Applying the transformation $(x, y) \to (-x, -y)$ to each point:

1. B: $(-2, 4) \to (2, -4)$
2. C: $(2, 6) \to (-2, -6)$
3. D: $(2, 8) \to (-2, -8)$
4. E: $(-2, 6) \to (2, -6)$

Conclusion:

The coordinates of the vertices after a 180° counterclockwise rotation around the origin are:

• B: $(2, -4)$
• C: $(-2, -6)$
• D: $(-2, -8)$
• E: $(2, -6)$

Would you like further details or have any questions?

Related Questions:

1. What would the coordinates be after a 90° counterclockwise rotation around the origin?
2. How would you find the coordinates after a 270° counterclockwise rotation around the origin?
3. How does rotating shapes affect their orientation and position on a graph?
4. How can you determine the center of rotation for any rotation in a coordinate plane?
5. What is the general formula for rotating a point around the origin by an arbitrary angle $\theta$?

Tip:

Remember that rotating a point 180° around the origin is the same as reflecting it through the origin.