Question

select the correct answer.
which sequence is not equivalent to the others?
○ a. a reflection across the y - axis, followed by a reflection across the x - axis, and then a 90° clockwise rotation about the origin
○ b. a 90° clockwise rotation about the origin and then a 180° rotation about the origin
○ c. a reflection across the x - axis, followed by a 90° counterclockwise rotation about the origin, and then a reflection across the x - axis
○ d. a 90° counterclockwise rotation about the origin

Explanation:

Step1: Analyze Option A

Let the point be \((x,y)\).

• Reflection across \(y\) - axis: \((x,y)\to(-x,y)\)
• Reflection across \(x\) - axis: \((-x,y)\to(-x, - y)\)
• \(90^{\circ}\) clockwise rotation: \((-x,-y)\to(-y,x)\)

Step2: Analyze Option B

• \(90^{\circ}\) clockwise rotation: \((x,y)\to(y,-x)\)
• \(180^{\circ}\) rotation: \((y,-x)\to(-y,x)\)

Step3: Analyze Option C

• Reflection across \(x\) - axis: \((x,y)\to(x,-y)\)
• \(90^{\circ}\) counter - clockwise rotation: \((x,-y)\to(y,x)\)
• Reflection across \(x\) - axis: \((y,x)\to(y,-x)\)

Step4: Analyze Option D

• \(90^{\circ}\) counter - clockwise rotation: \((x,y)\to(-y,x)\)

Answer:

C. a reflection across the \(x\) - axis, followed by a \(90^{\circ}\) counterclockwise rotation about the origin, and then a reflection across the \(x\) - axis