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Difficulty: Very hardTransformations in the Coordinate Plane

In the standard (x,y)(x, y) coordinate plane, a line segment ABAB has endpoints A(2,5)A(2, 5) and B(6,−1)B(6, -1). Segment ABAB is reflected across the line y=−xy = -x to form segment A′B′A'B'. Then, segment A′B′A'B' is dilated by a scale factor of 33 with a center of dilation at C(2,−1)C(2, -1) to form segment A′′B′′A''B''. What is the sum of the xx- and yy-coordinates of the midpoint of segment A′′B′′A''B''?

  1. A
    16
  2. B
    -8
  3. C
    -16
  4. D
    -18
  5. -20Answer

Answer

-20
To find the coordinates of the midpoint of the transformed segment A′′B′′A''B'', we can track the midpoint of the original segment ABAB through each transformation. The midpoint of ABAB is M(4,2)M(4, 2). Reflecting MM across the line y=−xy = -x swaps and negates its coordinates, resulting in M′(−2,−4)M'(-2, -4). Dilating M′M' by a scale factor of 33 centered at C(2,−1)C(2, -1) means finding the point M′′M'' such that the vector CM′′⃗\vec{CM''} is 33 times the vector CM′⃗\vec{CM'}. This is computed as M′′=(2+3(−2−2),−1+3(−4−(−1)))=(−10,−10)M'' = (2 + 3(-2 - 2), -1 + 3(-4 - (-1))) = (-10, -10). The sum of these coordinates is −10+(−10)=−20-10 + (-10) = -20.

Step-by-Step Solution

1
Find the midpoint of the original segment ABAB.
The midpoint MM of segment ABAB is M=(2+62,5+(−1)2)=(4,2)M = \left(\frac{2 + 6}{2}, \frac{5 + (-1)}{2}\right) = (4, 2).
Since rigid transformations (reflections) and similarity transformations (dilations) preserve midpoints, we can find the final midpoint by applying the sequence of transformations directly to the initial midpoint.
2
Reflect the midpoint M(4,2)M(4, 2) across the line y=−xy = -x to find the midpoint M′M' of segment A′B′A'B'.
Applying the reflection rule (x,y)→(−y,−x)(x, y) \rightarrow (-y, -x) to M(4,2)M(4, 2) yields M′(−2,−4)M'(-2, -4).
Reflecting a point across the line y=−xy = -x swaps its coordinates and negates both values.
3
Dilate the midpoint M′(−2,−4)M'(-2, -4) by a scale factor of 33 with the center of dilation at C(2,−1)C(2, -1) to find the midpoint M′′M'' of segment A′′B′′A''B''.
Using the dilation formula (x′′,y′′)=(xc+k(x′−xc),yc+k(y′−yc))(x'', y'') = (x_c + k(x' - x_c), y_c + k(y' - y_c)) with C(2,−1)C(2, -1) and k=3k = 3: x′′=2+3(−2−2)=−10x'' = 2 + 3(-2 - 2) = -10, and y′′=−1+3(−4−(−1))=−10y'' = -1 + 3(-4 - (-1)) = -10. This gives the midpoint M′′(−10,−10)M''(-10, -10).
A dilation centered at (xc,yc)(x_c, y_c) scales the horizontal and vertical distances from the center to the point by the scale factor kk.
4
Calculate the sum of the xx- and yy-coordinates of MM.
The sum of the coordinates is −10+(−10)=−20-10 + (-10) = -20.
The question asks for the sum of the coordinates of the final midpoint.

Key Concept

Applying composite transformations in the coordinate plane to geometric figures and midpoints.
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Topics

Transformations in the Coordinate Plane

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