Question

Triangle GFH has vertices G(2, –3), F(4, –1), and H(1, 1). The triangle is rotated 270° clockwise using the origin as the center of rotation. Which graph shows the rotated image? pick one answer

On a coordinate plane, triangle G prime H prime F prime has points (negative 3, negative 2), (1, negative 1), (negative 1, negative 4).
On a coordinate plane, triangle G prime H prime F prime has points (3, 2), (negative 1, 1), (1, 4).
On a coordinate plane, triangle G prime H prime F prime has points (2, negative 3), (1, 1), (4, negative 1).
On a coordinate plane, triangle G prime H prime F prime has points (2, 3), (4, 1), (1, negative 1).

Answer 1

To find the coordinates of triangle GFH after a 270° clockwise rotation around the origin, we can use the rotation transformation rules.

The transformation for a 270° clockwise rotation can be represented by the following transformation of the coordinates \( (x, y) \):

\[
(x, y) \rightarrow (y, -x)
\]

Now, let's apply this transformation to each vertex of triangle GFH:

1. Vertex G(2, -3):
\[
(2, -3) \rightarrow (-3, -2)
\]

2. Vertex F(4, -1):
\[
(4, -1) \rightarrow (-1, -4)
\]

3. Vertex H(1, 1):
\[
(1, 1) \rightarrow (1, -1)
\]

Thus, after the 270° clockwise rotation, the new coordinates for the vertices G', F', and H' are:
- G' = (-3, -2)
- F' = (-1, -4)
- H' = (1, -1)

Now let's check which answer choice matches these coordinates:
1. **On a coordinate plane, triangle G prime H prime F prime has points**: (negative 3, negative 2), (1, negative 1), (negative 1, negative 4) - **Not a match**.
2. **On a coordinate plane, triangle G prime H prime F prime has points**: (3, 2), (negative 1, 1), (1, 4) - **Not a match**.
3. **On a coordinate plane, triangle G prime H prime F prime has points**: (2, negative 3), (1, 1), (4, negative 1) - **Not a match**.
4. **On a coordinate plane, triangle G prime H prime F prime has points**: (2, 3), (4, 1), (1, negative 1) - **Not a match**.

After reviewing the original transformations, the correct rotated coordinates of the triangle GFH at a 270° clockwise rotation are not matching any options directly.

The expected points after rotation should be:
- G' = (-3, -2)
- F' = (-1, -4)
- H' = (1, -1)

From the provided answer choices, **the closest correct match based on the G’ and H’ points is:**
- Option 1: (G' = (-3, -2), H' = (1, -1), F' = (-1, -4))

However, the coordinates as presented in the options seem to have been presented differently. None of the direct listings are perfect matches without a re-gathering of clear points, but the information points towards the first alternative set since it fits G and H appropriately as derived through transformations recognized.

Thus, please ensure to have the adequate triangulation in representation as options weren't varying straight up with sequential correctness.