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 the image triangle U′V′W′ after rotating triangle UVW 90° counterclockwise, we can use the rotation transformation formula. The rule for rotating a point (x, y) 90° counterclockwise around the origin (0, 0) is given by the transformation:

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

Now, we will apply this transformation to each vertex of triangle UVW.

1. **For vertex U(−1, 1)**:
- x = −1, y = 1
- \( U' = (-1, 1) \to (−1, −1) \)

2. **For vertex V(0, −4)**:
- x = 0, y = −4
- \( V' = (0, -4) \to (4, 0) \)

3. **For vertex W(−4, −1)**:
- x = −4, y = −1
- \( W' = (-4, -1) \to (1, -4) \)

Now, we have the transformed coordinates:
- \( U'(-1, -1) \)
- \( V'(4, 0) \)
- \( W'(1, -4) \)

Now we check the options against these coordinates:

1. **Option 1:** \( U′(1, −1), V′(0, 4), W′(4, 1) \) - Incorrect
2. **Option 2:** \( U′(−1, −1), V′(4, 0), W′(1, −4) \) - Correct
3. **Option 3:** \( U′(−1, 1), V′(4, 0), W′(1, 4) \) - Incorrect
4. **Option 4:** \( U′(−1, 1), V′(0, −4), W′(−4, −1) \) - Incorrect

The correct answer is option 2, which has the vertices:
\[ U'(−1, −1), V'(4, 0), W'(1, −4) \]