Chapter 14.4, Problem 26WE

a.

To determine

To find: Distance invariant under $R_{O,90}$ .

Answer to Problem 26WE

Distance is invariant under a $R_{O,90}$ .

Explanation of Solution

Given:

  $R_{O,90}$ rotation.

Concept Used:

A rotation is a transformation suggested by rotating a paddle wheel. When wheel stops, new position of a paddle referred as the image of initial position.

Invariant a function, quantity, or property which remains unchanged when a specified transformation is applied.

Calculation:

As per the given problem

There is a $R_{O,90}$ rotation.

For reference,

  

From diagram it can be understand that distance between the points i.e. triangle side length remains same.

Hence,

Distance is invariant under a $R_{O,90}$ .

b.

To determine

To find: Angle measure invariant under $R_{O,90}$ .

Answer to Problem 26WE

Angle measure is invariant under a $R_{O,90}$ .

Explanation of Solution

Given:

  $R_{O,90}$ rotation

Concept Used:

A rotation is a transformation suggested by rotating a paddle wheel. When wheel stops, new position of a paddle referred as the image of initial position.

Invariant a function, quantity, or property which remains unchanged when a specified transformation is applied.

Calculation:

As per the given problem

There is a $R_{O,90}$ rotation.

For reference,

  

From diagram it can be understand that angle measure between the points i.e. triangle side angle remains same.

Hence,

Angle measure is invariant under a $R_{O,90}$ .

c.

To determine

To find: Area invariant under $R_{O,90}$ .

Answer to Problem 26WE

Area is invariant under a $R_{O,90}$ .

Explanation of Solution

Given:

  $R_{O,90}$ rotation

Concept Used:

A rotation is a transformation suggested by rotating a paddle wheel. When wheel stops, new position of a paddle referred as the image of initial position.

Invariant a function, quantity, or property which remains unchanged when a specified transformation is applied.

Calculation:

As per the given problem

There is a $R_{O,90}$ rotation.

For reference,

  

From diagram it can be understand that area between the points i.e. triangle area remains same.

Hence,

Area is invariant under a $R_{O,90}$ .

d.

To determine

To find: Orientation invariant under $R_{O,90}$ .

Answer to Problem 26WE

Orientation is not invariant under a $R_{O,90}$ .

Explanation of Solution

Given:

  $R_{O,90}$ rotation

Concept Used:

A rotation is a transformation suggested by rotating a paddle wheel. When wheel stops, new position of a paddle referred as the image of initial position.

Invariant a function, quantity, or property which remains unchanged when a specified transformation is applied.

Calculation:

As per the given problem

There is a $R_{O,90}$ rotation.

For reference,

  

From diagram it can be understand that orientation of shape changes i.e. triangle face opposite to the previous orientation.

Hence,

Orientation is not invariant under a $R_{O,90}$ .