Question

which of these transformations of abcd results in bc located in the same position?
a reflection across the x - axis followed by a reflection across the y - axis
a reflection across the x - axis followed by a translation of 4 upward.
a reflection across the y - axis followed by a dilation by a factor of 4.
a reflection across the y - axis followed by a translation of 2 to the left.

Explanation:

Step1: Analyze reflection across x - axis

Let the coordinates of the vertices of rectangle \(ABCD\) be \((x,y)\). A reflection across the \(x -\)axis changes the sign of the \(y -\)coordinate, so \((x,y)\to(x, - y)\).

Step2: Analyze reflection across y - axis after x - axis reflection

After the reflection across the \(x -\)axis, when we reflect across the \(y -\)axis, the sign of the \(x -\)coordinate changes. So \((x,-y)\to(-x, - y)\). If we consider the properties of the rectangle \(ABCD\) symmetric about the \(y -\)axis, a reflection across the \(x -\)axis followed by a reflection across the \(y -\)axis will map the rectangle back to a position where \(BC\) is in the same position as the original \(BC\) due to the symmetry of the rectangle about the \(y -\)axis.

Step3: Analyze other options

• For a reflection across the \(x -\)axis followed by a translation of 4 upward, the vertical position of \(BC\) will change.
• For a reflection across the \(y -\)axis followed by a dilation by a factor of 4, the size of the rectangle will change, so \(BC\) will not be in the same position.
• For a reflection across the \(y -\)axis followed by a translation of 2 to the left, the horizontal position of \(BC\) will change.

Answer:

A. A reflection across the \(x -\)axis followed by a reflection across the \(y -\)axis