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Difficulty: MediumTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, triangle PQRPQR has vertices P(2,1)P(2, 1), Q(5,1)Q(5, 1), and R(2,5)R(2, 5). The triangle is first rotated 90∘90^\circ counterclockwise about the origin to form triangle P′Q′R′P'Q'R'. Next, triangle P′Q′R′P'Q'R' is reflected across the yy-axis to form triangle P′′Q′′R′′P''Q''R''. What are the coordinates of the vertex R′′R''?

  1. (5,2)(5, 2)Answer
  2. B
    (−5,2)(-5, 2)
  3. C
    (5,−2)(5, -2)
  4. D
    (−5,−2)(-5, -2)
  5. E
    (2,−5)(2, -5)

Answer

(5,2)(5, 2)
The correct answer is the coordinate pair (5,2)(5, 2). First, rotating the point R(2,5)R(2, 5) counterclockwise by 90∘90^\circ about the origin uses the transformation rule (x,y)→(−y,x)(x, y) \rightarrow (-y, x), which maps R(2,5)R(2, 5) to R′(−5,2)R'(-5, 2). Next, reflecting the point R′(−5,2)R'(-5, 2) across the yy-axis uses the transformation rule (x,y)→(−x,y)(x, y) \rightarrow (-x, y), which maps R′(−5,2)R'(-5, 2) to R′′(5,2)R''(5, 2).

Step-by-Step Solution

1
Apply the rotation of 90∘90^\circ counterclockwise about the origin to the vertex R(2,5)R(2, 5).
The rule for a 90∘90^\circ counterclockwise rotation is (x,y)→(−y,x)(x, y) \rightarrow (-y, x). Applying this to R(2,5)R(2, 5) yields R′(−5,2)R'(-5, 2).
To find the coordinates after the first transformation step.
2
Apply the reflection across the yy-axis to the intermediate point R′(−5,2)R'(-5, 2).
The rule for reflection across the yy-axis is (x,y)→(−x,y)(x, y) \rightarrow (-x, y). Applying this to R′(−5,2)R'(-5, 2) yields R′′(5,2)R''(5, 2).
To find the final coordinates after the second transformation step.

Key Concept

Applying composite transformations (rotation followed by reflection) to coordinates in the coordinate plane.
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Topics

Transformations in the Coordinate Plane

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