To determine how to translate the rectangle in Quadrant III after rotating it 90 degrees clockwise about the origin, you need to find the translation vector that will align the rotated rectangle with the rectangle in Quadrant I.
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Determine the coordinates of the rectangles: Let's assume you have the bottom-left corner of the rectangle in Quadrant III at coordinates \((-x_1, -y_1)\) and the top-right corner at \((-x_2, -y_2)\). After you rotate it 90 degrees clockwise, the new coordinates of the corners will be \((y_1, -x_1)\) and \((y_2, -x_2)\).
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Identify the coordinates of the rectangle in Quadrant I: For the rectangle in Quadrant I, let's say its bottom-left corner is at \((x_a, y_a)\) and its top-right corner is at \((x_b, y_b)\).
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Calculate the translation vector: You need to translate the newly rotated rectangle so that its bottom-left corner aligns with the bottom-left corner of the rectangle in Quadrant I. Thus, the translation vector \((T_x, T_y)\) would be: \[ T_x = x_a - y_1 \quad \text{(horizontal translation)} \] \[ T_y = y_a - (-x_1) = y_a + x_1 \quad \text{(vertical translation)} \]
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Translate the rectangle: Apply this translation to all corners of the rotated rectangle to move it to the desired position in Quadrant I.
By following these steps, you will effectively translate the rotated rectangle to the correct position in Quadrant I, making it congruent to the original rectangle in Quadrant I.
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