Questions: Rectangle P'Q'R'S' is the image of rectangle PQRS after a translation followed by a rotation 270° counterclockwise about the origin. What was the translation? 8 units right and 2 units up 8 units left and 2 units down 2 units left and 8 units up 2 units right and 8 units down

Rectangle P'Q'R'S' is the image of rectangle PQRS after a translation followed by a rotation 270° counterclockwise about the origin. What was the translation? 8 units right and 2 units up 8 units left and 2 units down 2 units left and 8 units up 2 units right and 8 units down
Transcript text: Rectangle $P^{\prime} Q^{\prime} R^{\prime} S^{\prime}$ is the image of rectangle PQRS after a translation followed by a rotation $270^{\circ}$ counterclockwise about the origin. What was the translation? 8 units right and 2 units up 8 units left and 2 units down 2 units left and 8 units up 2 units right and 8 units down Submit Work it out Not feeling ready yet? These can help:
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Solution

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Solution Steps

Step 1: Reverse the rotation

To find the translation, we first need to reverse the rotation of 270° counterclockwise. This is equivalent to rotating 90° clockwise. When rotating a point (x, y) 90° clockwise around the origin, the new point becomes (y, -x). Applying this to P'(7, 8), Q'(4, 4), R'(3, 5), and S'(4, 9), we get P''(8, -7), Q''(4, -4), R''(5, -3), and S''(9, -4).

Step 2: Compare corresponding vertices

Now, compare the coordinates of the original vertices P(-6,-1), Q(-3,-4), R(-2,-6), and S(-5,-3) with the coordinates of the rotated vertices P''(8,-7), Q''(4,-4), R''(5,-3), and S''(9,-4).

Step 3: Calculate the translation

Focus on one pair of corresponding vertices, such as P(-6, -1) and P''(8, -7). The horizontal translation is 8 - (-6) = 14 units right. The vertical translation is -7 - (-1) = -6 units, or 6 units down. Let's verify this with another pair of vertices, such as S(-5, -3) and S''(9, -4). Horizontal translation: 9 - (-5) = 14. Vertical translation: -4 - (-3) = -1. This does not confirm the same translation, so we need to recheck our work. It seems like a rotation of 270° counterclockwise about the origin would transform point (x, y) to (-y, x). Applying this to rectangle PQRS, we get P''(1, -6), Q''(4, -3), R''(6, -2), S''(3, -5). Comparing S(-5, -3) and S''(3, -5), we get the translation +8 in x direction and -2 in y direction.

Final Answer: The translation is 8 units right and 2 units down.

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