Question

which rigid transformation would map △aqr to △akp? a rotation about point a a reflection across the line containing (overline{ar}) a reflection across the line containing (overline{aq}) a rotation about point r

Explanation:

Step1: Analyze rotation

Rotation about a point changes the orientation of a figure. But here, the key is to check the congruence and the axis of transformation.

Step2: Analyze reflection

Reflection across a line. If we consider the line containing \(\overline{AR}\), we can see that \(\triangle AQR\) and \(\triangle AKP\) are mirror - images with respect to the line \(AR\). Because \(AQ = AK\) (marked as equal in the figure), \(AR=AR\) (common side) and \(\angle QAR=\angle KAP\) (given by the angle markings). A reflection across the line \(AR\) will map \(Q\) to \(K\) and \(R\) to \(R\) (since \(R\) lies on the line of reflection), thus mapping \(\triangle AQR\) to \(\triangle AKP\).
Rotation about \(A\) or \(R\) won't give the exact mapping as per the congruence shown (by side - angle - side as \(AQ = AK\), \(\angle QAR=\angle KAP\), \(AR = AP\)) and reflection across \(AQ\) is incorrect as the mapping of points doesn't match.

Answer:

a reflection across the line containing \(\overline{AR}\)