Question

7, in the diagram below, congruent triangles pen and pal are drawn. which rigid motion maps △pen onto △pal? 1) a point reflection of △pen through p 2) a reflection of △pen over the angle bisector of ∠epa 3) a rotation of △pen about point p, mapping \\( \overline { p e } \\) onto \\( \overline { p a } \\) 4) a translation of △pen along \\( \overline { e a } \\), mapping point e onto a

Explanation:

Step1: Analyze point reflection

A point reflection through \(P\) would not map \(\triangle PEN\) to \(\triangle PAL\) as the orientation and position - relationship in the diagram do not match a point - reflection property.

Step2: Analyze reflection over angle bisector

There is no indication in the problem - given diagram that the line between the two triangles is the angle bisector of \(\angle EPA\).

Step3: Analyze rotation

Since \(\triangle PEN\cong\triangle PAL\), a rotation about point \(P\) (a rigid motion that preserves shape and size) can map \(\overline{PE}\) onto \(\overline{PA}\) (because \(PE = PA\) as the triangles are congruent), and the other corresponding parts (\(PN = PL\) and \(EN=AL\)) will also be mapped correctly.

Step4: Analyze translation

A translation along \(\overline{EA}\) mapping \(E\) to \(A\) would change the position of the triangle in a non - rotational way. But from the congruence and the common vertex \(P\), a translation does not account for the rotational - like mapping of the two triangles about \(P\).

Answer:

1. a rotation of \(\triangle PEN\) about point \(P\), mapping \(\overline{PE}\) onto \(\overline{PA}\)