Question

Triangle UVW is drawn with vertices at U(−1, 1), V(0, −4), W(−4, −1). Determine the coordinates of the vertices for the image, triangle U′V′W′, if the preimage is rotated 90° counterclockwise.

U′(1, −1), V′(0, 4), W′(4, 1)
U′(−1, −1), V′(4, 0), W′(1, −4)
U′(−1, 1), V′(4, 0), W′(1, 4)
U′(−1, 1), V′(0, −4), W′(−4, −1)

Answer 1

To find the coordinates of the vertices for triangle U'V'W' after a 90° counterclockwise rotation about the origin, we can use the rotation formula:

If a point \( (x, y) \) is rotated 90° counterclockwise, the new coordinates \( (x', y') \) can be calculated using:

\[
(x', y') = (-y, x)
\]

Let's apply this to each vertex of triangle UVW:

1. **Vertex U at (-1, 1)**:
\[
U' = (-1, 1) \rightarrow (1, -1)
\]

2. **Vertex V at (0, -4)**:
\[
V' = (0, -4) \rightarrow (-4, 0)
\]

3. **Vertex W at (-4, -1)**:
\[
W' = (-4, -1) \rightarrow (1, -4)
\]

Now we summarize the rotated vertices:
- \( U' (1, -1) \)
- \( V' (-4, 0) \)
- \( W' (1, -4) \)

Thus, the correct listing for triangle U'V'W' is \( U'(1, -1), V'(-4, 0), W'(1, -4) \).

However, looking at the options provided, there seems to be an inconsistency with one of the vertex coordinates:

None of the given options match with the calculated points. As per the calculation:
- \( U' \) should be \( (1, -1) \)
- \( V' \) should be \( (-4, 0) \)
- \( W' \) should be \( (1, -4) \)

Make sure you verify against options. If any option closely resembles, pick that; otherwise, it may need checking from the source of the options.