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Difficulty: MediumTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, point PP is rotated 90∘90^\circ counterclockwise about the origin, and then translated 33 units to the left and 55 units down. If the coordinates of the resulting image point P′P' are (−6,2)(-6, 2), what are the coordinates of the original point PP?

  1. A
    (−3,9)(-3, 9)
  2. B
    (−7,−3)(-7, -3)
  3. C
    (7,−3)(7, -3)
  4. (7,3)(7, 3)Answer
  5. E
    (3,−9)(3, -9)

Answer

(7,3)(7, 3)
The correct answer is (7,3)(7, 3). Working backward from the final image point (−6,2)(-6, 2), we first reverse the translation of 33 units left and 55 units down by translating the point 33 units right and 55 units up, yielding (−6+3,2+5)=(−3,7)(-6+3, 2+5) = (-3, 7). We then reverse the 90∘90^\circ counterclockwise rotation by performing a 90∘90^\circ clockwise rotation about the origin. The rule for a 90∘90^\circ clockwise rotation is (x,y)→(y,−x)(x, y) \rightarrow (y, -x), which maps (−3,7)(-3, 7) to (7,3)(7, 3).

Step-by-Step Solution

1
Identify the sequence of transformations and set up the backward process.
The forward process is: Point P→P \rightarrow Rotate 90∘90^\circ counterclockwise →\rightarrow Translate left 33 and down 5→P′(−6,2)5 \rightarrow P'(-6, 2). To find the original point PP, we must reverse these steps starting from P′P': Translate right 33 and up 5→5 \rightarrow Rotate 90∘90^\circ clockwise →\rightarrow Point PP.
Working backward requires reversing both the order and the direction of each transformation in the sequence.
2
Apply the reverse translation to the image point P′(−6,2)P'(-6, 2).
Translating 33 units to the right adds 33 to the xx-coordinate: −6+3=−3-6 + 3 = -3. Translating 55 units up adds 55 to the yy-coordinate: 2+5=72 + 5 = 7. This gives the intermediate point P1(−3,7)P_1(-3, 7).
Reversing a translation of 33 units left and 55 units down requires adding 33 to the xx-coordinate and 55 to the yy-coordinate.
3
Apply the reverse rotation to the intermediate point P1(−3,7)P_1(-3, 7).
The reverse of a 90∘90^\circ counterclockwise rotation is a 90∘90^\circ clockwise rotation. The coordinate rule for a 90∘90^\circ clockwise rotation about the origin is (x,y)→(y,−x)(x, y) \rightarrow (y, -x). Applying this rule to P1(−3,7)P_1(-3, 7) yields (7,−(−3))=(7,3)(7, -(-3)) = (7, 3).
Rotating a point 90∘90^\circ clockwise about the origin swaps the coordinates and negates the new yy-coordinate.

Key Concept

Reversing composite transformations in the coordinate plane
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Topics

Transformations in the Coordinate Plane

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