Question

in each part, rotate the solid figure about the origin exactly onto the dashed figure.
if this is not possible, click on
ot possible\.

Explanation:

Step1: Analyze the first pair of figures

For the first set of figures (the parallelograms), when we rotate a figure about the origin, the distance of each point from the origin remains the same. The solid - colored parallelogram and the dashed - colored parallelogram have the same shape and size. We can observe that a rotation of \(180^{\circ}\) about the origin will map the solid - colored parallelogram onto the dashed - colored parallelogram. Using the rotation rule \((x,y)\to(-x, - y)\) for a \(180^{\circ}\) rotation about the origin, if we take a vertex of the solid - colored parallelogram, say \((x_1,y_1)\), after a \(180^{\circ}\) rotation about the origin, it will be mapped to \((-x_1,-y_1)\) which is a vertex of the dashed - colored parallelogram.

Step2: Analyze the second pair of figures

For the second set of figures (the trapezoids), the solid - colored trapezoid and the dashed - colored trapezoid have different orientations that cannot be achieved by a rotation about the origin. A rotation about the origin is a transformation that preserves the shape and the relative position of points with respect to the origin. If we consider the properties of rotation (such as the angle of rotation \(\theta\) where \(0^{\circ}<\theta<360^{\circ}\)), there is no angle \(\theta\) for which the solid - colored trapezoid can be rotated about the origin to exactly match the dashed - colored trapezoid.

Answer:

For the first pair of figures (parallelograms): Rotate \(180^{\circ}\) about the origin.
For the second pair of figures (trapezoids): Not possible.