Chapter 14.8, Problem 2E

a.

To determine

To make a list of four symmetries of the rhombus shown.

Answer to Problem 2E

Following are the four symmetries of the rhombus shown in figure.

1. Identity symmetry.

2. Symmetry with respect to reflection on line k .

3. Symmetry with respect to reflection on line j .

4. The half turn symmetry that is point symmetry about O .

Explanation of Solution

Given information:

The given statement is that list the four symmetries of the rhombus shown.

Following are the four symmetries of the rhombus shown in figure.

5. Identity symmetry.

6. Symmetry with respect to reflection on line k .

7. Symmetry with respect to reflection on line j .

8. The half turn symmetry that is point symmetry about O .

b.

To determine

To make a group table showing all products of two symmetries.

Answer to Problem 2E

The table of all products of two symmetries.

$°$	I $R_j$ $R_k$ $H_o$
I  $R_j$ $R_k$ $H_o$	I $R_j$ $R_k$ $H_o$ $R_j$ I $H_o$ $R_k$ $R_k$ $H_o$ I $R_j$ $H_o$ $R_k$ $R_j$ I

Explanation of Solution

Given information:

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

The table of all products of two symmetries.

$°$	I $R_j$ $R_k$ $H_o$
I  $R_j$ $R_k$ $H_o$	I $R_j$ $R_k$ $H_o$ $R_j$ I $H_o$ $R_k$ $R_k$ $H_o$ I $R_j$ $H_o$ $R_k$ $R_j$ I

Where $R_j$ represents the reflectionof figure with respect to linej .

And $R_k$ represents the reflectionof figure with respect to linek .

  $H_o$ Represents half turn symmetry.

I is the identity symmetry.

c.

To determine

To write if the table in part b identical to the table of symmetries for the rectangle.

Answer to Problem 2E

Yes, the table in part b is identical to the table of symmetries for the rectangle.

Explanation of Solution

Given information:

The given statement is that if the table in part b identical to the table of symmetries for the rectangle.

Yes, the table in part b is identical to the table of symmetries for the rectangle. Since the rectangle has four symmetries and this rhombus has also four symmetries. And all the symmetries of the rectangle are self-inverse similarly all the symmetries of the rhombus is also self-inverse. Hence, the symmetry table is identical.