Question

the graph shows triangles pqr and vwx. is pqr similar to vwx? justify your answer. yes, because a rotation 90° clockwise around the origin followed by a reflection over the x - axis maps pqr onto vwx. yes, because a rotation 90° clockwise around the origin followed by a reflection over the y - axis maps pqr onto vwx. no, because ∠r and ∠x do not have the same measure.

Explanation:

Step1: Analyze the transformation

Rotation and reflection are rigid transformations. If a rotation \(90^{\circ}\) clockwise around the origin followed by a reflection over the \(x -\)axis maps \(\triangle PQR\) onto \(\triangle VWX\), then \(\triangle PQR\) and \(\triangle VWX\) are congruent. Congruent triangles are similar (with a scale factor of \(1\)).

• For a rotation \(90^{\circ}\) clockwise around the origin, the transformation rule is \((x,y)\to(y, - x)\).
• For a reflection over the \(x -\)axis, the transformation rule is \((x,y)\to(x,-y)\).

Step2: Check the other options

• If we consider a rotation \(90^{\circ}\) clockwise around the origin followed by a reflection over the \(y -\)axis:
• Rotation \(90^{\circ}\) clockwise: \((x,y)\to(y, - x)\)
• Reflection over \(y -\)axis: \((x,y)\to(-x,y)\). This will not map \(\triangle PQR\) onto \(\triangle VWX\) as per the graph.
• For the statement “No, because \(\angle R\) and \(\angle X\) do not have the same measure”:
• Since a rotation and a reflection (rigid transformations) preserve angle measures. If \(\triangle PQR\) can be mapped onto \(\triangle VWX\) by rigid transformations, \(\angle R\) and \(\angle X\) must have the same measure.

Answer:

Yes, because a rotation \(90^{\circ}\) clockwise around the origin followed by a reflection over the \(x -\)axis maps \(PQR\) onto \(VWX\).