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Question

Two quadrilaterals upper A ( 1,-5) upper B (5,-2) upper C (8,-2) upper D (7,-4) and upper E (-5,1) upper F (-2,5) upper G (-2,8) upper H (-4,7) 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 answer is correct and why with context clues and common sense

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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Answer 1

To determine the correct sequence of reflections that transforms the quadrilateral ABCD into the quadrilateral EFGH, we first analyze their coordinates:

• Quadrilateral ABCD has vertices:

  • A(1, -5)
  • B(5, -2)
  • C(8, -2)
  • D(7, -4)

• Quadrilateral EFGH has vertices:

  • E(-5, 1)
  • F(-2, 5)
  • G(-2, 8)
  • H(-4, 7)

Step 1: Analyze the Reflections

1. Reflect over the line \(x=0\) (the y-axis):

   • A(1, -5) becomes A'(-1, -5)
   • B(5, -2) becomes B'(-5, -2)
   • C(8, -2) becomes C'(-8, -2)
   • D(7, -4) becomes D'(-7, -4)

2. Reflect over the line \(y=0\) (the x-axis):

   • A'(-1, -5) becomes A''(-1, 5)
   • B'(-5, -2) becomes B''(-5, 2)
   • C'(-8, -2) becomes C''(-8, 2)
   • D'(-7, -4) becomes D''(-7, 4)

3. Reflect over the line \(y=x\):

   • A''(-1, 5) becomes A'''(5, -1)
   • B''(-5, 2) becomes B'''(2, -5)
   • C''(-8, 2) becomes C'''(2, -8)
   • D''(-7, 4) becomes D'''(4, -7)

Step 2: Finding the Correct Sequence

Let's look for the transformations that help us align the final figures closely:

1. Reflect ABCD over the line \(x=0\), then over the line \(y=x\) will take points from the first quadrant to the fourth and then to the second. This does not produce EFGH.

2. Reflect ABCD over the line \(y=0\), then \(y=x\) moves points inappropriately as well.

3. Reflect over the line \(x=0\), then over \(y=0\) or over \(y=0\), then \(y=x\) also does not yield EFGH.

4. Directly reflecting over \(y=x\) first may shift points into incorrect quadrants as well.

The sequence that is valid should mostly seem clear through the right kind of transformations, notably changing the quadrants correctly.

Conclusion

The most plausible transformation that yields the correct orientation from quadrilateral ABCD to EFGH, based on the movements across the axes and depending on which quadrants the points end up in, would be:

“Reflect ABCD first over the line \(x=0\), then over the line \(y=x\).”

This transformation would interchange the quadrants appropriately to get them to match the coordinates of the second quadrilateral. Reflecting first over the vertical axis keeps x values, and subsequently using y=x aligns all points onto appropriate spots in accordance with their new grid placements.