Chapter 14.3, Problem 11WE

a.

To determine

To find:Whether distance is invariant under translation.

Answer to Problem 11WE

Yes, distance is invariant under translation.

Explanation of Solution

Given:

  $T:(x,y)\to (x\text{'},y\text{'})$

Concept Used:

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.An invariant is a property of a mathematical object which remains unchanged, after operations or transformations of a certain type are applied to the objects.

Calculation:

As per the given problem

Translation $T:(x,y)\to (x\text{'},y\text{'})$

Distance between points is invariant under translation.

i.e. $\begin{array}{l}AB=A\text{'}B\text{'}\\ BC=B\text{'}C\text{'}\\ CA=C\text{'}A\text{'}\end{array}$

Hence,

Distance is invariant under a translation.

b.

To determine

To find:Whether angle measure is invariant under translation.

Answer to Problem 11WE

Yes, angle measure is invariant under translation.

Explanation of Solution

Given:

  $T:(x,y)\to (x\text{'},y\text{'})$

Concept Used:

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.An invariant is a property of a mathematical object which remains unchanged, after operations or transformations of a certain type are applied to the objects.

Calculation:

As per the given problem

Translation $T:(x,y)\to (x\text{'},y\text{'})$

Angle measure between points is invariant under translation.

i.e. $\begin{array}{l}\angle ABC=\angle A\text{'}B\text{'}C\text{'}\\ \angle BCA=\angle B\text{'}C\text{'}A\text{'}\\ \angle CAB=\angle C\text{'}A\text{'}B\text{'}\end{array}$

Hence,

Angle measure is invariant under a translation.

c.

To determine

To find:Whether area is invariant under translation.

Answer to Problem 11WE

Yes, area is invariant under translation.

Explanation of Solution

Given:

  $T:(x,y)\to (x\text{'},y\text{'})$

Concept Used:

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.An invariant is a property of a mathematical object which remains unchanged, after operations or transformations of a certain type are applied to the objects.

Calculation:

As per the given problem

Translation $T:(x,y)\to (x\text{'},y\text{'})$

Area of polygon formed by points is invariant under translation.

i.e. All the corresponding sides are equivalent & angle formed at all the points are same.

Hence,

Area is invariant under a translation.

d.

To determine

To find:Whether orientation is invariant under translation.

Answer to Problem 11WE

Yes, orientation is invariant under translation.

Explanation of Solution

Given:

  $T:(x,y)\to (x\text{'},y\text{'})$

Concept Used:

A translation moves ("slides") an object a fixed distance in a given direction without changing its size or shape, and without turning it or flipping it.The original object is called the pre-image, and the translation is called the image.An invariant is a property of a mathematical object which remains unchanged, after operations or transformations of a certain type are applied to the objects.Orientation is how the relative pieces of an object are arranged.

Calculation:

As per the given problem

Translation $T:(x,y)\to (x\text{'},y\text{'})$

Slope of $PQ$ remain same as for $P\text{'}Q\text{'}$ .

Hence,

Orientation is invariant under a translation.