Question

Parallelogram RSTU is rotated 45° clockwise using the origin as the center of rotation.

On a coordinate plane, parallelogram R S T U has points (2, 3), (5, 3), (7, 1), (4, 1).

Which graph shows the image of RSTU?
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (negative 1, 4), (1, 6), (4, 6), (2, 4).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.5, 1), (5.5, negative 1), (5.5, negative 4), (3.5, negative 2).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (2, negative 3), (5, negative 3), (7, negative 1), (4, negative 1).
On a coordinate plane, parallelogram R prime S prime T prime U prime has points (3.2, 4.8), (5.5, 2.5), (5.5, negative 0.5), (3.2, 1.8).

Answer 1

Rotate (x,y) by 45° clockwise (θ = −45°):
x' = (x+y)·(√2/2), y' = (y−x)·(√2/2).

Apply to the vertices R(2,3), S(5,3), T(7,1), U(4,1):

- R' = ((2+3)√2/2, (3−2)√2/2) = (5√2/2, √2/2) ≈ (3.54, 0.71)
- S' = ((5+3)√2/2, (3−5)√2/2) = (4√2, −√2) ≈ (5.66, −1.41)
- T' = ((7+1)√2/2, (1−7)√2/2) = (4√2, −3√2) ≈ (5.66, −4.24)
- U' = ((4+1)√2/2, (1−4)√2/2) = (5√2/2, −3√2/2) ≈ (3.54, −2.12)

These rounded coordinates match the second graph: (3.5, 1), (5.5, −1), (5.5, −4), (3.5, −2). So the second graph is the image.