To determine the length of the segment after it is rotated 270° clockwise, we first need to find the length of the segment itself and then analyze how rotation affects it.
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Find the length of the segment with endpoints (0, 4) and (3, 4): The segment lies horizontally at \(y = 4\), spanning from \(x = 0\) to \(x = 3\). The length of the segment can be calculated as: \[ Length = |x_2 - x_1| = |3 - 0| = 3 \]
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Rotation of the segment 270° clockwise: Rotating a point (x, y) 270° clockwise around the origin follows this transformation: \[ (x', y') = (y, -x) \]
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For the first endpoint (0, 4): \[ (0', 4') = (4, -0) = (4, 0) \]
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For the second endpoint (3, 4): \[ (3', 4') = (4, -3) \]
After the rotation, the new endpoints are (4, 0) and (4, -3).
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Calculating the length of the new segment: The new segment has endpoints (4, 0) and (4, -3). This segment is vertical at \(x = 4\) and spans from \(y = 0\) to \(y = -3\). The length can be calculated as: \[ Length' = |y_2 - y_1| = |-3 - 0| = |-3| = 3 \]
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Conclusion: The length of the segment after rotating it 270° clockwise remains the same. Therefore, the length of the segment after the rotation is: \[ \boxed{3} \]
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