Question

On a coordinate plane, 2 squares are shown. The first square has points A (1, 1), B (4, 1), C (4, 4), D (1, 4). The second square has points A prime (negative 1, negative 4), B prime (negative 4, negative 4), C prime (negative 4, negative 1), D prime (negative 1, negative 1).
Micaela tried to rotate the square 180° about the origin. Is her rotation correct? If not, explain why.

No, she translated the figure instead of rotating it.
No, she reflected the figure instead of rotating it.
No, the vertices of the image and pre-image do not correspond.
Yes, the rotation is correct.

Answer 1

To determine if Micaela's rotation of the square is correct or not, we first need to calculate the coordinates of the points after performing a 180° rotation about the origin.

When a point \((x, y)\) is rotated 180° around the origin, its image will be \((-x, -y)\).

Let's apply this transformation to the points of the first square:

- Point A (1, 1):
\[
(-1, -1)
\]

- Point B (4, 1):
\[
(-4, -1)
\]

- Point C (4, 4):
\[
(-4, -4)
\]

- Point D (1, 4):
\[
(-1, -4)
\]

Now we compare these points with the coordinates of square A'B'C'D':
- A' = (-1, -4)
- B' = (-4, -4)
- C' = (-4, -1)
- D' = (-1, -1)

From the results, we can see that the points obtained after the rotation do not match the points of the second square A'B'C'D'. Therefore, Micaela's 180° rotation is incorrect.

The correct answer is:
**No, the vertices of the image and pre-image do not correspond.**