To determine the coordinates of point A'' after a rotation of 270 degrees counter-clockwise about the origin and a reflection across the line \(y=4\), we can follow these steps:
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Rotate point A(5,3) by 270 degrees counter-clockwise about the origin: The rotation formula for a point \((x, y)\) counter-clockwise by an angle \(\theta\) is given by: \[ (x', y') = (x \cdot \cos(\theta) - y \cdot \sin(\theta), x \cdot \sin(\theta) + y \cdot \cos(\theta)) \] For a rotation of 270 degrees (which is equivalent to -90 degrees), the cosine and sine values are: \[ \cos(270^\circ) = 0, \quad \sin(270^\circ) = -1 \]
Plugging in \(A(5,3)\): \[ A' = (5 \cdot 0 - 3 \cdot -1, 5 \cdot -1 + 3 \cdot 0) = (0 + 3, -5 + 0) = (3, -5) \]
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Reflect point A'(3,-5) across the line \(y=4\): To reflect a point \((x, y)\) across a horizontal line \(y=k\), the formula is: \[ (x, y') = (x, k - (y - k)) = (x, 2k - y) \]
Here, \(k = 4\): \[ A'' = (3, 2 \cdot 4 - (-5)) = (3, 8 + 5) = (3, 13) \]
Thus, the coordinates of point A'' are \((3, 13)\).
The correct response is: (3, 13).