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Difficulty: MediumTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, the line segment PQPQ has endpoints P(−3,5)P(-3, 5) and Q(1,−1)Q(1, -1). Segment PQPQ is reflected across the line y=−xy = -x, and the resulting image is then translated 33 units to the right and 22 units down to form segment P′Q′P'Q'. What is the midpoint of segment P′Q′P'Q'?

  1. A
    (−5,3)(-5, 3)
  2. B
    (0,−2)(0, -2)
  3. (1,−1)(1, -1)Answer
  4. D
    (−5,−1)(-5, -1)
  5. E
    (1,3)(1, 3)

Answer

(1,−1)(1, -1)
The correct answer is the coordinate point (1,−1)(1, -1). First, the midpoint of the original segment PQPQ is found using the midpoint formula: (−3+12,5+(−1)2)=(−1,2)\left( \frac{-3 + 1}{2}, \frac{5 + (-1)}{2} \right) = (-1, 2). Reflecting this midpoint across the line y=−xy = -x swaps and negates the coordinates, transforming (−1,2)(-1, 2) to (−2,1)(-2, 1). Finally, translating this point 33 units to the right and 22 units down results in the coordinates (−2+3,1−2)=(1,−1)(-2 + 3, 1 - 2) = (1, -1).

Step-by-Step Solution

1
Find the midpoint of the original segment PQPQ.
The midpoint is M(−1,2)M(-1, 2).
Since the midpoint is invariant under linear coordinate transformations, we can find the midpoint of the original segment and transform it directly rather than transforming both endpoints and finding the new midpoint.
2
Reflect the midpoint across the line y=−xy = -x.
The reflected point is M1(−2,1)M_1(-2, 1).
Reflecting a point across the line y=−xy = -x maps the coordinate (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x).
3
Translate the reflected point 33 units to the right and 22 units down.
The final point is M′(1,−1)M'(1, -1).
Translating a point 33 units right adds 33 to the xx-coordinate, and translating 22 units down subtracts 22 from the yy-coordinate: (−2+3,1−2)=(1,−1)(-2 + 3, 1 - 2) = (1, -1).

Key Concept

Transformations in the Coordinate Plane (Reflection and Translation)

Alternative Method

Instead of transforming the midpoint directly, you can apply the transformations to the endpoints P(−3,5)P(-3, 5) and Q(1,−1)Q(1, -1) individually. Reflecting both across the line y=−xy = -x gives P1(−5,3)P_1(-5, 3) and Q1(1,−1)Q_1(1, -1). Translating both points 33 units right and 22 units down gives P′(−2,1)P'(-2, 1) and Q′(4,−3)Q'(4, -3). The midpoint of P′Q′P'Q' is then calculated as (−2+42,1+(−3)2)=(1,−1)\left( \frac{-2 + 4}{2}, \frac{1 + (-3)}{2} \right) = (1, -1).
Estimated Time:1m 30s
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Topics

Transformations in the Coordinate Plane

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