Chapter 14.6, Problem 15WE

a.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 15WE

The properties which do not change under the composite of a rotation and a translation are

  $a,b,c\text{ and }d.$

Explanation of Solution

Given a composite of rotation and translation.

Calculation:

Now consider a composite of rotation and transformation. According to the definition of rotation,

a point rotation about a point therefore, it does not bring any changes in the distance, angle measure, area and orientation.

Similarly , a translation only glides a point from one place to another ,it also doesn’t change distance, angle measure, area and orientation.

Now ,according to the isometries theorem ,the composite of two isometries is an isometry.

Therefore ,distance ,angle measure ,area and orientation are invariant under the composite of

rotation and translation .

Hence , the properties which do not change under the composite of a rotation and translation are

  $a,b,c\text{ and }d.$

b.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 15WE

The properties which do not change under the composite of a rotation and a translation are

  $a,b,c\text{ and }d.$

Explanation of Solution

Given a composite of rotation and translation.

Calculation:

Now consider a composite of rotation and transformation. According to the definition of rotation,

a point rotation about a point therefore, it does not bring any changes in the distance, angle measure, area and orientation.

Similarly , a translation only glides a point from one place to another ,it also doesn’t change distance, angle measure, area and orientation.

Now ,according to the isometries theorem ,the composite of two isometries is an isometry.

Therefore ,distance ,angle measure ,area and orientation are invariant under the composite of

rotation and translation .

Hence , the properties which do not change under the composite of a rotation and translation are

  $a,b,c\text{ and }d.$

c.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 15WE

The properties which do not change under the composite of a rotation and a translation are

  $a,b,c\text{ and }d.$

Explanation of Solution

Given a composite of rotation and translation.

Calculation:

Now consider a composite of rotation and transformation. According to the definition of rotation,

a point rotation about a point therefore, it does not bring any changes in the distance, angle measure, area and orientation.

Similarly , a translation only glides a point from one place to another ,it also doesn’t change distance, angle measure, area and orientation.

Now ,according to the isometries theorem ,the composite of two isometries is an isometry.

Therefore ,distance ,angle measure ,area and orientation are invariant under the composite of

rotation and translation .

Hence , the properties which do not change under the composite of a rotation and translation are

  $a,b,c\text{ and }d.$