Question

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

Explanation:

Step1: Analyze reflection across the x - axis

When a point \((x,y)\) is reflected across the \(x\) - axis, the transformation is \((x,y)\to(x, - y)\). Let the vertices of rectangle \(PQRS\) be \(P(x_1,y_1)\), \(Q(x_1,y_2)\), \(R(x_2,y_2)\), \(S(x_2,y_1)\). After reflection across the \(x\) - axis, the vertices of \(P'Q'R'S'\) are \(P'(x_1,-y_1)\), \(Q'(x_1,-y_2)\), \(R'(x_2,-y_2)\), \(S'(x_2,-y_1)\)

Step2: Analyze option C

• Reflection across the \(y\) - axis: The transformation for a point \((x,y)\) is \((x,y)\to(-x,y)\). After reflection of \(P'Q'R'S'\) across the \(y\) - axis, the vertices become \(P''(-x_1,-y_1)\), \(Q''(-x_1,-y_2)\), \(R''(-x_2,-y_2)\), \(S''(-x_2,-y_1)\)
• Reflection across the \(x\) - axis: The transformation for a point \((x,y)\) is \((x,y)\to(x, - y)\). After reflection of \(P''Q''R''S''\) across the \(x\) - axis, the vertices are \(P'''(-x_1,y_1)\), \(Q'''(-x_1,y_2)\), \(R'''(-x_2,y_2)\), \(S'''(-x_2,y_1)\) (not correct)

Step3: Analyze option D

• Reflection across the \(y\) - axis: The transformation for a point \((x,y)\) is \((x,y)\to(-x,y)\). After reflection of \(P'Q'R'S'\) across the \(y\) - axis, the vertices are \(P''(-x_1,-y_1)\), \(Q''(-x_1,-y_2)\), \(R''(-x_2,-y_2)\), \(S''(-x_2,-y_1)\)
• \(180^{\circ}\) rotation around the origin: The transformation for a point \((x,y)\) is \((x,y)\to(-x,-y)\). After \(180^{\circ}\) rotation of \(P''Q''R''S''\) around the origin, the vertices are \(P'''(x_1,y_1)\), \(Q'''(x_1,y_2)\), \(R'''(x_2,y_2)\), \(S'''(x_2,y_1)\) (correct)

Answer:

D. reflection across the y - axis followed by a 180 degree rotation around the origin