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 rotate triangle UVW with vertices at U(−1, 1), V(0, −4), W(−4, −1) by 90° counterclockwise around the origin, we can use the following transformation for the coordinates (x, y):

When rotated by 90° counterclockwise, the new coordinates (x', y') are given by:
\[
x' = -y
\]
\[
y' = x
\]

Now, let's calculate the coordinates for each vertex of triangle UVW.

1. For U(−1, 1):
- \(x' = -y = -1\)
- \(y' = x = -1\)
- So, U' = (−1, −1)

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

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

Thus, the coordinates of the vertices for triangle U′V′W′ after the rotation are:
U′(−1, −1), V′(4, 0), W′(1, −4).

Matching this with the options provided, we can see that the correct answer is:
U′(−1, −1), V′(4, 0), W′(1, −4).