Chapter 14.6, Problem 16WE

a.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 16WE

The properties which are invariant under the composite of two dilation are $b\text{ and }d.$

Explanation of Solution

Given a composite of reflection and dilation.

Calculation:

Check whether the distance varies under a composite of reflection and a dilation or not.

As the composite of two isometries is an isometry, but under dilation , the distance does depend on the scalar factor.

Thus the distance varies under the composite of reflection and dilation.

b.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 16WE

The properties which are invariant under the composite of two dilation are $b\text{ and }d.$

Explanation of Solution

Given a composite of reflection and dilation.

Calculation:

Check whether the angle measure varies under a composite of reflection and a dilation or not .

As the composite of two isometries is an isometry, and transformations, dilation and reflection,

The angle measure does not changes therefore, angle measure is invariant under the composite of reflection and dilation.

c.

To determine

To identify the invariant under the composite of reflection.

Answer to Problem 16WE

The properties which are invariant under the composite of two dilation are $b\text{ and }d.$

Explanation of Solution

Given a composite of reflection and dilation.

Calculation:

Check whether the area varies under a composite of reflection and a dilation or not .

As the composite of two isometries is an isometry, but under dilation, the area does depend on

The scalar factor therefore, area varies under the composite of reflection and dilation.