To graph the image of rectangle CDEF after a 270° clockwise rotation around the origin, you first need the coordinates of the vertices of the rectangle. Let's assume the coordinates of rectangle CDEF are as follows:
- C(x1, y1)
- D(x2, y2)
- E(x3, y3)
- F(x4, y4)
The formula for rotating a point (x, y) around the origin by an angle θ is:
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For a counterclockwise rotation: ( (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) )
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For a clockwise rotation, you can use: ( (x', y') = (x \cos \theta + y \sin \theta, -x \sin \theta + y \cos \theta) )
Since a 270° clockwise rotation is equivalent to a 90° counterclockwise rotation, we can use:
\[ \begin{align*} x' &= x \cos(90°) - y \sin(90°) = 0 - y = -y \ y' &= x \sin(90°) + y \cos(90°) = x + 0 = x \end{align*} \]
Thus, after a 270° clockwise rotation, the new coordinates will be:
\[ (x', y') = (-y, x) \]
To find the new coordinates for each vertex of rectangle CDEF, apply the transformation to each vertex:
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For point C(x1, y1):
- \( C' = (-y1, x1) \)
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For point D(x2, y2):
- \( D' = (-y2, x2) \)
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For point E(x3, y3):
- \( E' = (-y3, x3) \)
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For point F(x4, y4):
- \( F' = (-y4, x4) \)
Plot these new points \( C', D', E', F' \) to graph the image of CDEF after the rotation.
If you provide specific coordinates for points C, D, E, and F, I can compute the exact coordinates for the rotated points.