Question

1. here are 2 polygons:

select all sequences of translations, rotations, and reflections below that would take polygon p to
polygon q.
a. rotate 180° around point a.
b. rotate 60° counterclockwise around point a
and then reflect over the line fa.
c. translate so that a is taken to j. then reflect
over line ba.
d. reflect over line ba and then translate by
directed line segment ba.
e. reflect over the line ba and then rotate 60°
counterclockwise around point a.

Explanation:

Step1: Analyze option a

Rotating \(180^{\circ}\) around point \(A\). If we consider the properties of rotation, a \(180^{\circ}\) rotation is a central symmetry. For polygon \(P\) and \(Q\), a \(180^{\circ}\) rotation around point \(A\) will map polygon \(P\) to polygon \(Q\) as the relative positions of the vertices with respect to point \(A\) are reversed in a \(180^{\circ}\) turn.

Step2: Analyze option b

Rotate \(60^{\circ}\) counter - clockwise around point \(A\) and then reflect over the line \(FA\). First, the \(60^{\circ}\) rotation changes the orientation of the polygon with respect to point \(A\). Then, the reflection over line \(FA\) (a line of symmetry in the context of the geometric transformation) will map the rotated polygon (from the \(60^{\circ}\) rotation) to polygon \(Q\).

Step3: Analyze option c

Translate so that \(A\) is taken to \(J\). Then reflect over line \(BA\). The translation moves the polygon so that vertex \(A\) (of polygon \(P\)) coincides with vertex \(J\) (in the target position). Then, the reflection over line \(BA\) (which acts as a mirror for the translated polygon) maps it to polygon \(Q\).

Step4: Analyze option d

Reflect over line \(BA\) and then translate by directed line segment \(BA\). The reflection over line \(BA\) flips the polygon with respect to line \(BA\). Then, the translation by the directed line segment \(BA\) (which has a specific length and direction) does not map polygon \(P\) to polygon \(Q\) as the combined effect of this reflection - translation does not align the vertices of \(P\) with those of \(Q\) correctly.

Step5: Analyze option e

Reflect over the line \(BA\) and then rotate \(60^{\circ}\) counterclockwise around point \(A\). The reflection over line \(BA\) changes the position of the polygon with respect to line \(BA\). Then, the \(60^{\circ}\) rotation around point \(A\) does not result in polygon \(P\) being mapped to polygon \(Q\) as the sequence of operations does not account for the correct spatial transformation.

Answer:

A. Rotate \(180^{\circ}\) around point \(A\), B. Rotate \(60^{\circ}\) counterclockwise around point \(A\) and then reflect over the line \(FA\), C. Translate so that \(A\) is taken to \(J\). Then reflect over line \(BA\)