Question

determine a series of transformations that would map polygon abcd onto polygon abcd. a rotation of 90° counterclockwise about the origin followed by a dilation centered at the origin with a scale factor of.

Explanation:

Step1: Analyze rotation

The orientation of polygon \(ABCD\) has changed to match \(A'B'C'D'\) in a way that suggests a \(90^{\circ}\) counter - clockwise rotation about the origin. The general rule for a \(90^{\circ}\) counter - clockwise rotation about the origin for a point \((x,y)\) is \((x,y)\to(-y,x)\).

Step2: Analyze dilation

After rotation, we need to find the scale factor of dilation. Let's assume a vertex of \(ABCD\) say \(A(x_1,y_1)\) and its corresponding vertex \(A'(x_2,y_2)\) after rotation and dilation. By comparing the distances of corresponding points from the origin or the lengths of corresponding sides, we find that the scale factor \(k=\frac{1}{2}\). For example, if we consider the distance of a point from the origin before and after dilation, if the distance of a point \(P\) from the origin is \(d_1\) and after dilation is \(d_2\), and \(d_2=\frac{1}{2}d_1\), the scale factor of dilation centered at the origin is \(\frac{1}{2}\).

Answer:

A rotation of \(90^{\circ}\) counterclockwise about the origin followed by a dilation centered at the origin with a scale factor of \(\frac{1}{2}\).