Question

which transformation will carry the rhombus onto itself? a reflection across the line y = 2 a reflection across the line y = x a reflection across the x - axis a rotation 90° counterclockwise about the origin

Explanation:

Step1: Recall properties of rhombus

A rhombus has two - types of symmetry: line - symmetry and rotational symmetry.

Step2: Analyze line of symmetry

The given rhombus is symmetric about the x - axis. When we reflect a point \((x,y)\) across the x - axis, the transformation rule is \((x,y)\to(x, - y)\). For the rhombus, points above the x - axis will map to corresponding points below the x - axis and vice - versa in a way that the rhombus is carried onto itself.

Step3: Analyze other options

• Reflection across \(y = 2\): The rhombus is not symmetric about \(y=2\), so this will not carry it onto itself.
• Reflection across \(y = x\): The rhombus is not symmetric about \(y = x\), so this will not carry it onto itself.
• Rotation \(90^{\circ}\) counter - clockwise about the origin: A \(90^{\circ}\) counter - clockwise rotation about the origin using the rule \((x,y)\to(-y,x)\) will not map the rhombus onto itself.

Answer:

a reflection across the x - axis