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Difficulty: HardTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, line segment PQPQ has endpoints P(−2,3)P(-2, 3) and Q(4,1)Q(4, 1). First, segment PQPQ is rotated 90∘90^\circ counterclockwise about the origin to form segment P′Q′P'Q'. Next, segment P′Q′P'Q' is reflected across the line y=xy = x to form segment P′′Q′′P''Q''. What are the coordinates of the midpoint of segment P′′Q′′P''Q''?

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

Answer

The coordinates of the midpoint of segment P′′Q′′P''Q'' are (1,−2)(1, -2).
The midpoint of the original segment PQPQ is (1,2)(1, 2). Under a 90∘90^\circ counterclockwise rotation about the origin, the point (x,y)(x, y) maps to (−y,x)(-y, x), so (1,2)(1, 2) maps to (−2,1)(-2, 1). Under a reflection across the line y=xy = x, the point (x,y)(x, y) maps to (y,x)(y, x), so (−2,1)(-2, 1) maps to (1,−2)(1, -2). Since rigid transformations preserve midpoints, the midpoint of the final segment is (1,−2)(1, -2).

Step-by-Step Solution

1
Calculate the midpoint of the original segment PQPQ.
Midpoint M=(−2+42,3+12)=(1,2)M = \left(\frac{-2+4}{2}, \frac{3+1}{2}\right) = (1, 2)
Since rotation and reflection are rigid transformations (isometries), the midpoint of the transformed segment is the transformed midpoint of the original segment.
2
Apply a 90∘90^\circ counterclockwise rotation about the origin to the midpoint coordinate (1,2)(1, 2).
Intermediate midpoint M′=(−2,1)M' = (-2, 1)
The coordinate rule for a 90∘90^\circ counterclockwise rotation about the origin is (x,y)→(−y,x)(x, y) \rightarrow (-y, x).
3
Apply a reflection across the line y=xy = x to the intermediate midpoint (−2,1)(-2, 1).
Final midpoint M′′=(1,−2)M'' = (1, -2)
The coordinate rule for reflection across the line y=xy = x is (x,y)→(y,x)(x, y) \rightarrow (y, x).

Key Concept

Applying composite transformations (rotations and reflections) to geometric figures on the coordinate plane, utilizing the property that the midpoint of a transformed segment is the transformed midpoint of the original segment.
Estimated Time:1m 30s
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Topics

Transformations in the Coordinate Plane

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