The figure below is dilated with the center of dilation at the origin and a scale factor of \frac{1}{2}
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and then rotated 180° counterclockwise. What are the coordinates of the image of point E after these transformations?
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To find the coordinates of the image of point E after these transformations, we need to follow two steps: dilation and rotation.
1. Dilation:
The dilation is performed with a center at the origin and a scale factor of \(\frac{1}{2}\). This means that the image of point E is half the distance from the origin as the original point E.
To dilate point E, we multiply the x and y coordinates of E by the scale factor \(\frac{1}{2}\).
Let's assume the coordinates of point E are (x, y), then the dilated coordinates of E, let's call it E', can be found using the following formulas:
x' = \(\frac{1}{2}\) * x
y' = \(\frac{1}{2}\) * y
2. Rotation:
After dilation, the image of point E', which we obtained in the previous step, is rotated 180° counterclockwise.
To rotate a point 180° counterclockwise, we take the negative of the x and y coordinates.
The final coordinates of the image of point E after these transformations can be found using the following formulas:
x_final = -x'
y_final = -y'
Substituting the dilated coordinates of E' into these formulas, we get:
x_final = -\(\frac{1}{2}\) * x
y_final = -\(\frac{1}{2}\) * y
Therefore, the coordinates of the image of point E after these transformations are (-\(\frac{1}{2}\) * x, -\(\frac{1}{2}\) * y).