To find the length of the segment after it is rotated 270° clockwise, we can follow these steps:
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Identify the endpoints of the segment: The segment has endpoints at (0, 4) and (3, 4).
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Calculate the length of the segment: The length of the segment can be calculated using the distance formula, but in this case, we can observe that both points have the same y-coordinate (4), meaning it’s a horizontal line segment. The length can be calculated as follows:
\[ \text{Length} = |x_2 - x_1| = |3 - 0| = 3 \]
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Rotation of the segment: Now we will rotate the endpoints 270° clockwise about the origin (0, 0). The rotation of a point (x, y) by an angle θ (in degrees) around the origin can be calculated using the transformation: \[ (x', y') = (x \cos \theta - y \sin \theta, x \sin \theta + y \cos \theta) \] For a rotation of 270° clockwise, \(\cos(270°) = 0\) and \(\sin(270°) = -1\). Therefore, the transformation becomes: \[ (x', y') = (y, -x) \]
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For point (0, 4): \[ (x', y') = (4, -0) = (4, 0) \]
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For point (3, 4): \[ (x', y') = (4, -3) \]
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New endpoints after rotation: The new endpoints of the segment after rotation will be (4, 0) and (4, -3).
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Calculate the length of the new segment: The length of the new segment is however calculated using the distance formula:
\[ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} = \sqrt{(4 - 4)^2 + (-3 - 0)^2} = \sqrt{0 + 9} = 3 \]
Thus, the length of the segment after it is rotated 270° clockwise remains 3.
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