Chapter 9.4, Problem 43HP

To determine

To find: Whether the order of the rotation is sometimes, always or never affect the location of the final image.

Answer to Problem 43HP

The order of the rotation affects the location of the final image

Explanation of Solution

Given:

The two rotations are performed on the single image.

Calculation:

Consider the case when the rotation is performed on the single image the order of the rotation is not affected by the final image when the two rotation are centred at the same points.

Consider the example that $\Delta ABC$ is rotated $45°$ clockwise about the origin and then rotate $60°$ clockwise about the origin then, $\Delta A^{″}{B}^{″}{C^{\prime }}^{\prime }$ is same as the figure rotated to $60°$ clockwise about the origin and then it is rotated $45°$ clockwise about the origin.

Consider the figure shown below in Figure 1

  

Figure 1

Consider that triangle ABC is rotate $45°$ clockwise about the origin and then rotated $60°$ clockwise about $P\left(2,3\right)$ . The triangle $\Delta A^{″}{B}^{″}{C^{\prime }}^{\prime }$ is different than the figure that is rotated $60°$ clockwise about $P\left(2,3\right)$ and then rotated $45°$ about the clockwise about the origin.

The required graph is shown in Figure 2

  

Figure

Thus, the order of the rotation affects the location of the final image