Write the coordinates of the vertices after a rotation 270° counterclockwise around the origin.
(x,y)→(y,-x)
try it with an easy point, like (1,0) on the x-axis.
repeat it to show that it works.
To rotate a point counterclockwise around the origin, we can apply the following rotation formula:
(x', y') = (x*cosθ - y*sinθ, x*sinθ + y*cosθ)
Let's assume we have four vertices A, B, C, and D with their respective coordinates:
Vertex A: (x1, y1)
Vertex B: (x2, y2)
Vertex C: (x3, y3)
Vertex D: (x4, y4)
To find the new coordinates after rotating 270° counterclockwise, we need to substitute each vertex's coordinates in the rotation formula:
New coordinates of A:
(x1', y1') = (x1 * cos270° - y1 * sin270°, x1 * sin270° + y1 * cos270°)
New coordinates of B:
(x2', y2') = (x2 * cos270° - y2 * sin270°, x2 * sin270° + y2 * cos270°)
New coordinates of C:
(x3', y3') = (x3 * cos270° - y3 * sin270°, x3 * sin270° + y3 * cos270°)
New coordinates of D:
(x4', y4') = (x4 * cos270° - y4 * sin270°, x4 * sin270° + y4 * cos270°)
Remember that cos270° = 0 and sin270° = -1. So, substituting the values, we get:
New coordinates of A:
(x1', y1') = (x1 * 0 - y1 * -1, x1 * -1 + y1 * 0)
(x1', y1') = (y1, -x1)
New coordinates of B:
(x2', y2') = (x2 * 0 - y2 * -1, x2 * -1 + y2 * 0)
(x2', y2') = (y2, -x2)
New coordinates of C:
(x3', y3') = (x3 * 0 - y3 * -1, x3 * -1 + y3 * 0)
(x3', y3') = (y3, -x3)
New coordinates of D:
(x4', y4') = (x4 * 0 - y4 * -1, x4 * -1 + y4 * 0)
(x4', y4') = (y4, -x4)
Therefore, after rotating 270° counterclockwise around the origin, the new coordinates of the vertices are:
Vertex A': (y1, -x1)
Vertex B': (y2, -x2)
Vertex C': (y3, -x3)
Vertex D': (y4, -x4)