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Difficulty: HardTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a line segment has endpoints A(−3,2)A(-3, 2) and B(1,4)B(1, 4). The segment is reflected across the line y=−xy = -x to form segment A′B′A'B', which is then translated 55 units to the right and 33 units down to form segment A′′B′′A''B''. What are the coordinates of the midpoint of segment A′′B′′A''B''?

  1. (2,−2)(2, -2)Answer
  2. B
    (−8,4)(-8, 4)
  3. C
    (8,−4)(8, -4)
  4. D
    (0,−4)(0, -4)
  5. E
    (6,0)(6, 0)

Answer

(2,−2)(2, -2)
The correct answer is (2,−2)(2, -2). The midpoint of the original segment ABAB is calculated as M(−1,3)M(-1, 3). Reflecting MM across the line y=−xy = -x yields M′(−3,1)M'(-3, 1). Translating M′M' by 55 units right and 33 units down results in the final midpoint M′′(2,−2)M''(2, -2).

Step-by-Step Solution

1
Find the coordinates of the midpoint of the original segment ABAB.
The midpoint MM of ABAB is calculated using the midpoint formula: M=(−3+12,2+42)=(−1,3)M = \left(\frac{-3 + 1}{2}, \frac{2 + 4}{2}\right) = (-1, 3).
Since translations, reflections, and dilations preserve midpoints, the midpoint of the final image segment A′′B′′A''B'' is the same point obtained by applying the transformations to the midpoint of the original segment ABAB.
2
Apply the reflection across the line y=−xy = -x to the midpoint M(−1,3)M(-1, 3).
Using the reflection rule (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x), the reflected midpoint is M′=(−3,1)M' = (-3, 1).
A reflection across y=−xy = -x swaps the coordinates of a point and negates both values.
3
Apply the translation of 55 units right and 33 units down to the point M′(−3,1)M'(-3, 1).
Using the translation rule (x,y)→(x+5,y−3)(x, y) \rightarrow (x + 5, y - 3), the final midpoint is M′′=(−3+5,1−3)=(2,−2)M'' = (-3 + 5, 1 - 3) = (2, -2).
Translating a point right increases the x-coordinate, and translating it down decreases the y-coordinate.

Key Concept

Applying composite transformations (reflection across y=−xy = -x followed by translation) to geometric objects and midpoints in the coordinate plane.
Estimated Time:2m 0s
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Topics

Transformations in the Coordinate Plane

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