Chapter 14.8, Problem 5E

a.

To determine

To write the number of symmetries which are self-inverse out of four symmetries in previous exercise.

Answer to Problem 5E

There are two symmetries which are self-inverse.

Explanation of Solution

Given information:

A transformation that is its own inverse is called a self-inverse.

There are two symmetries which are self-inverse. And the symmetry which are self-inverse are one at the identity and one at the rotation of $180°$ . It can be better understand by the symmetry table.

$°$	I $R_{0,}$ ${}_{90}$ $R_{0,}$ ${}_{180}$ $R_{0,}$ ${}_{270}$
I $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$	I $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ $R_0$ ${}_{90}$ $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{180}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{120}$ $R_0$ ${}_{270}$ I $R_0$ ${}_{90}$ $R_{0,}$ ${}_{270}$

b.

To determine

To write the number of symmetries which areself-inverse out of four symmetries of the rectangle.

Answer to Problem 5E

All four symmetries of the rectangle are self-inverse

Explanation of Solution

Given information:

A transformation that is its own inverse is called a self-inverse.

All four symmetries of the rectangle are self-inverse because in each symmetry the figure reflects onto itself.