Question

rectangle pqrs is reflected across the x - axis. the image is pqrs. which two further transformations will carry the image back onto pqrs?
a. reflection across the y - axis followed by a reflection across the x - axis
b. reflection across the y - axis followed by a 180 degree rotation around the origin
c. reflection across the y - axis followed by a reflection across the line with the equation y = - x
d. reflection across the line with equation y = x followed by a 180 degree rotation around the origin

Explanation:

Step1: Analyze reflection across y - axis and x - axis

When a point \((x,y)\) is reflected across the \(y\) - axis, the transformation is \((x,y)\to(-x,y)\). When it is reflected across the \(x\) - axis, the transformation is \((x,y)\to(x, - y)\).

Step2: Analyze reflection across \(y=-x\)

The transformation for reflection across the line \(y =-x\) is \((x,y)\to(-y,-x)\).

Step3: Analyze rotation of \(180^{\circ}\)

The transformation for a \(180^{\circ}\) rotation about the origin is \((x,y)\to(-x,-y)\).

Step4: Analyze reflection across \(y = x\)

The transformation for reflection across the line \(y=x\) is \((x,y)\to(y,x)\).

Step5: Combine transformations

Let's assume a general point \(P(x_1,y_1)\) in the first - quadrant.

• For option A: First reflection across \(y\) - axis \((x_1,y_1)\to(-x_1,y_1)\), then reflection across \(x\) - axis \((-x_1,y_1)\to(-x_1,-y_1)\).
• For option B: First reflection across \(y\) - axis \((x_1,y_1)\to(-x_1,y_1)\), then \(180^{\circ}\) rotation \((-x_1,y_1)\to(x_1,-y_1)\).
• For option C: First reflection across \(y\) - axis \((x_1,y_1)\to(-x_1,y_1)\), then reflection across \(y=-x\) \((-x_1,y_1)\to(-y_1,x_1)\).
• For option D: First reflection across \(y = x\) \((x_1,y_1)\to(y_1,x_1)\), then \(180^{\circ}\) rotation \((y_1,x_1)\to(-y_1,-x_1)\).

If we first reflect a rectangle in the first quadrant across the \(x\) - axis (mapping \((x,y)\to(x,-y)\)), and then we want to get it back.
A reflection across the \(y\) - axis followed by a reflection across \(y=-x\) will not map it back. A reflection across \(y\) - axis followed by \(180^{\circ}\) rotation will not map it back. A reflection across \(y=x\) followed by \(180^{\circ}\) rotation will not map it back.
If we first reflect a point \((x,y)\) across the \(x\) - axis to get \((x,-y)\), then reflecting across the \(y\) - axis gives \((-x,-y)\), and then a \(180^{\circ}\) rotation about the origin (transformation \((-x,-y)\to(x,y)\)) will map the rectangle back.

Answer:

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