Question

select all the sequences of transformations that could take figure p to figure q.
a single reflection
a single rotation
a single translation
a translation, then a reflection
a reflection, then a different reflection

Explanation:

Step1: Analyze single reflection

A single reflection can map Figure \(P\) to Figure \(Q\). For example, reflecting over the line \(y = -x\) can transform Figure \(P\) to Figure \(Q\).

Step2: Analyze single rotation

A single rotation (a \(180^{\circ}\) rotation about the point \((-0.5,-1)\) for example) can map Figure \(P\) to Figure \(Q\).

Step3: Analyze single translation

A single translation cannot change the orientation of the figure. Since Figure \(P\) and Figure \(Q\) have different orientations (in terms of which side is on which coordinate - direction), a single translation is not sufficient.

Step4: Analyze translation then reflection

We can first translate Figure \(P\) (for example, translate it \(5\) units to the right and \(2\) units down) and then reflect it (over a vertical or horizontal line) to get Figure \(Q\).

Step5: Analyze reflection then different reflection

Two reflections (composite of reflections) can be equivalent to a rotation. For example, reflecting over a vertical line and then over a horizontal line (or vice - versa) can map Figure \(P\) to Figure \(Q\).

Answer:

a single reflection, a single rotation, a translation, then a reflection, a reflection, then a different reflection