A quadrilateral with points A(1,4), B(5,4), C(4,3), and D(2,2) is reflected about the y-axis to form quadrilateral A'B'C'D'. Which of these sequences of transformations will also carry the quadrilateral ABCD onto A'B'C'D'? A. Rotating the quadrilateral 90 degrees clockwise about the origin and then reflecting it across the x-axis B. Rotating the quadrilateral 180 degrees about the origin and then reflecting it across the x-axis C. Rotating the quadrilateral 90 degrees counterclockwise about the origin and then reflecting it across the line y=x D. Rotating the quadrilateral 180 degrees about the origin and then reflecting it across the line y=x

Expert Answers

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We are asked to find a transformation from the list that is equivalent, in the sense that it maps the given quadrilateral to the same image, as a reflection over the y-axis. We are given quadrilateral ABCD with A(1,4), B(5,4), C(4,3), and D(2,2).

Note that a reflection over the y-axis maps each point (x,y) in the plane to (-x,y). (This can be written as `R_y:(x,y)->(-x,y)` .) Thus we know the image of the transformation: A'(-1,4), B'(-5,4), C'(-4,3), and D'(-2,2).

B is the correct answer.

A composition of transformations is a sequence of transformations performed from right to left. Compositions are, in general, not commutative, so order matters.

Consider the alternatives:

A. `R_x @ R_(O,-90^@)` A clockwise rotation of 90 degrees about the origin (note the negative sign) followed by a reflection over the x-axis. This composition maps points from the first quadrant to the first quadrant (e.g., A(1,4) gets sent to (4,-1) by the rotation and then to (4,1) by the reflection). This is equivalent to reflecting over the line y=x.

C. `R_(y=x)@R(O,90^@)` A counterclockwise rotation of 90 degrees about the origin followed by a reflection over the line y=x. This composition sends points from the first quadrant to the fourth quadrant (e.g., A(1,4) gets mapped to (-4,1) by the rotation and then to (1,-4) by the reflection). This is equivalent to a reflection over the x-axis.

D. `R_(y=x)@R_(O,180^@)` A rotation about the origin of 180 degrees (or a half-turn about the origin) followed by a reflection over the line y=x. This composition maps points from the first quadrant to the third quadrant (e.g., A(1,4) gets mapped to (-1,-4) by the rotation and then to (-4,-1) by the reflection).

B is the only composition given that maps first quadrant points to the second quadrant. `R_x @ R_(O,180^@)` is a half-turn followed by a reflection over the x-axis (e.g., A(1,4) gets sent to (-1,-4) by the rotation and then to (-1,4) by the reflection). You should check the other points.

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