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.

Question

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: \[ U'(−1, −1), V'(4, 0), W'(1, −4) \]