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Question

Two quadrilaterals upper A upper B upper C upper D and upper E upper F upper G upper H are marked in the second and fourth quadrants of a coordinate plane respectively.

What reflections or sequence of reflections transforms figure ABCD into figure EFGH?

(1 point)
Responses which is the true correct answer a b c or d and tell me by solving it why it is that one

Reflect ABCD first over the line x=0
, then over the line y=x.
Reflect  cap A cap b cap c cap d  first over the line x is equal to 0, then over the line  y is equal to x .

Reflect ABCD first over the line y=0, then over the line y=x. 
Reflect  cap A cap b cap c cap d  first over the line  y is equal to 0 , then over the line  y is equal to x .

Reflect ABCD first over the line x=0, then over the line y=0.
Reflect  cap A cap b cap c cap d  first over the line  x is equal to 0 , then over the line  y is equal to 0 .

Reflect ABCD over the line y=x
Reflect  cap A cap b cap c cap d  over the line  y is equal to x
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GPT-4o mini · Answer

To determine which reflections transform quadrilateral ABCD into quadrilateral EFGH, we need to look at the properties of reflections in a coordinate plane.

Assume:

Quadrilateral ABCD is located in the second quadrant (where x < 0 and y > 0).
Quadrilateral EFGH is located in the fourth quadrant (where x > 0 and y < 0).

To go from the second quadrant to the fourth quadrant, the following transformations should occur:

Reflect over the line y=0 (the x-axis): This reflection will take points in the second quadrant to the third quadrant. For example, a point (x, y) will be reflected to (x, -y). So a point in the second quadrant will move to the third quadrant.
Reflect over the line x=0 (the y-axis): This reflection will take points from the third quadrant to the fourth quadrant. For example, a point (x, y) will be reflected to (-x, y), flipping the x-coordinate to negative.
Reflect over the line y=x: This transformation will swap the x and y coordinates of the shape, but in our case, the quadrilateral is already in place after the first two transformations.

Given these necessary reflections, we can see:

The transformation sequence that reflects over the line y = 0 first would move ABCD to the third quadrant.
The subsequent reflection over the line x = 0 would then move it to the fourth quadrant.

Now let's compare the given options:

Option B

Reflect ABCD first over the line y=0, then over the line y=x.

In this option, the first reflection correctly takes ABCD to the third quadrant (which is not what we want) but does not help to get to EFGH.

Option A

Reflect ABCD first over the line x=0, then over the line y=x.

This will not work because the first reflection takes the shape to the left instead of down.

Option C

Reflect ABCD first over the line x=0, then over the line y=0.

This option does not correctly move the shape to the desired position either.

Option D

Reflect ABCD over the line y=x.

This single reflection won't be sufficient to achieve the transformation needed.

Based on the analysis:

The correct transformations are achieved by reflecting ABCD first over y=0 (the x-axis), then over x=0 (the y-axis). This will correctly position ABCD transformed into EFGH.

Thus, the correct response is: Reflect ABCD first over the line y=0, then over the line y=x.