Question

△abc underwent a sequence of rigid transformations to give △abc. which transformations might have taken place?
a. a reflection across the y - axis followed by a reflection across the x - axis
b. a rotation 90° clockwise about the origin followed by a reflection across the x - axis
c. a rotation 270° clockwise about the origin followed by a reflection across the x - axis
d. a reflection across the x - axis followed by a reflection across the y - axis

Explanation:

Step1: Analyze Option A

• Reflection across \(y\) - axis: \((x,y)\to(-x,y)\). Then reflection across \(x\) - axis: \((-x,y)\to(-x, - y)\).
• Let's assume a point \(C(-1,2)\) (approximate from the graph). After reflection across \(y\) - axis: \((1,2)\), then after reflection across \(x\) - axis: \((1,-2)\) (matches \(C'\) in the graph).
• Let's check the general property. Two reflections (first \(y\), then \(x\)) is equivalent to a rotation of \(180^{\circ}\) about the origin. But we can also check by the rule of composition of reflections.

Step2: Analyze Option B

• Rotation \(90^{\circ}\) clockwise about the origin: \((x,y)\to(y,-x)\). Then reflection across \(x\) - axis: \((y,-x)\to(y,x)\). This does not match the transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).

Step3: Analyze Option C

• Rotation \(270^{\circ}\) clockwise about the origin: \((x,y)\to(-y,x)\). Then reflection across \(x\) - axis: \((-y,x)\to(-y,-x)\). This does not match the transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).

Step4: Analyze Option D

• Reflection across \(x\) - axis: \((x,y)\to(x,-y)\). Then reflection across \(y\) - axis: \((x,-y)\to(-x,-y)\). This does not match the transformation from \(\triangle ABC\) to \(\triangle A'B'C'\).

Answer:

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