Question

determine a series of transformations that would map polygon ( abcde ) onto polygon ( abcde ).

Explanation:

Step1: Analyze the rotation

First, consider a rotation of \(180^{\circ}\) counter - clockwise about the origin. The rule for a \(180^{\circ}\) counter - clockwise rotation about the origin is \((x,y)\to(-x, -y)\).

Step2: Analyze the reflection

After a \(180^{\circ}\) rotation, we observe that a reflection over the \(x -\)axis (rule \((x,y)\to(x,-y)\)) is not needed. But if we first consider a \(90^{\circ}\) clockwise rotation (equivalent to a \(270^{\circ}\) counter - clockwise rotation, rule \((x,y)\to(y,-x)\)) and then a reflection over the \(x -\)axis (rule \((x,y)\to(x,-y)\)).

Alternatively, we can also think in terms of visual inspection. If we rotate polygon \(ABCDE\) \(90^{\circ}\) clockwise (or \(270^{\circ}\) counter - clockwise) about the origin, and then reflect it over the \(x -\)axis.

Let's take a point as an example. Suppose a point \(A(x,y)\) on polygon \(ABCDE\). After a \(90^{\circ}\) clockwise rotation about the origin, it becomes \((y, -x)\). Then after a reflection over the \(x -\)axis, it becomes \((y,x)\) which is consistent with the mapping to the corresponding point \(A'\) on \(A'B'C'D'E'\)

Answer:

A rotation of \(270^{\circ}\) counterclockwise about the origin followed by a reflection over the \(x -\)axis.