Question

Parallelogram ABCD is rotated 45° counterclockwise using the origin as the center of rotation.

On a coordinate plane, parallelogram A B C D has points (2, 5), (2, 2), (negative 1, negative 1), (negative 1, 2).

Which graph shows the image of ABCD?
On a coordinate plane, parallelogram A prime B prime C prime D prime has points (negative 2, 5), (0, 3), (0, negative 1.2), (negative 2, 1).
On a coordinate plane, parallelogram A prime B prime C prime D prime has points (negative 2, 5), (negative 2, 1), (0, negative 1), (0, 3).
On a coordinate plane, parallelogram A prime B prime C prime D prime has points (5, 2), (3, 0), (negative 1.5, 0), (0.5, 2).
On a coordinate plane, parallelogram A prime B prime C prime D prime has points (5, 2), (1, 2), (negative 1.5, 0), (2.8, 0).

Answer 1

Rotation by 45° CCW about the origin: x' = (x - y)/√2, y' = (x + y)/√2.

Compute for each vertex:
- A(2,5) → A' = ((2−5)/√2, (2+5)/√2) = (−3/√2, 7/√2) ≈ (−2.12, 4.95) → (−2, 5)
- B(2,2) → B' = (0, 4/√2) = (0, 2.83) → (0, 3)
- C(−1,−1) → C' = (0, −2/√2) = (0, −1.41) → (0, −1.2 ≈ −1.41)
- D(−1,2) → D' = (−3/√2, 1/√2) ≈ (−2.12, 0.71) → (−2, 1)

These match the first listed graph: A'(−2,5), B'(0,3), C'(0,−1.2), D'(−2,1).