Chapter 14.4, Problem 33WE

a.

To determine

To draw: Coordinate grid with origin $O$ along points $A(0,3)\&B(4,1)$ .

Answer to Problem 33WE

Coordinate grid with origin $O$ along points $A(0,3)\&B(4,1)$ is plotted.

Explanation of Solution

Given:

Coordinate $A(0,3)\&B(4,1)$ .

Concept Used:

2D cartesian coordinate grid where coordinates are represented as $(x,y)$ .

Calculation:

As per the given problem

  

Hence,

Coordinate grid with origin $O$ along points $A(0,3)\&B(4,1)$ is plotted.

b.

To determine

To find: To plot image of $A\&B$ by $R_{O,270}$ .

Answer to Problem 33WE

  $A\text{'}$ & $B\text{'}$ image of $A$ & $B$ plotted.

Explanation of Solution

Given:

Coordinate $A(0,3)\&B(4,1)$ .

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.

Calculation:

As per the given problem

Coordinate $A(0,3)\&B(4,1)$

  $\begin{array}{l}{R}_{O,270}=R_{O,270-360}=R_{O,-90}\\ R_{O,-90}:(x,y)\to (y,-x)\end{array}$

Rotation about point $O$ clockwise ${90}^o$ .

  $\begin{array}{l}{R}_{O,90}(A):(0,3)\to (3,0)=A\text{'}\\ R_{O,90}(B):(4,1)\to (1,-4)=B\text{'}\end{array}$

  

Hence,

  $A\text{'}$ & $B\text{'}$ image of $A$ & $B$ plotted.

c.

To determine

To compare: Slopes of $\overleftrightarrow{AB}\&\overleftrightarrow{A\text{'}B\text{'}}$

Answer to Problem 33WE

Both line $\overleftrightarrow{AB}\&\overleftrightarrow{A\text{'}B\text{'}}$ are perpendicular.

Explanation of Solution

Given:

Coordinate $A(0,3)\&B(4,1)$ .

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.

Calculation:

As per the given problem

  $A(0,3)\&B(4,1)$ & their image $A\text{'}(3,0)\&B\text{'}(1,-4)$ .

Slope: -

  $\begin{array}{l}\overleftrightarrow{AB}=\frac{3-1}{0-4}=\frac{2}{-4}=\frac{-1}{2}\\ \overleftrightarrow{A\text{'}B\text{'}}=\frac{-4-0}{1-3}=\frac{-4}{-2}=2\end{array}$

  $m(\overleftrightarrow{AB})*m(\overleftrightarrow{A\text{'}B\text{'}})=-1$

Product of slopes shows that both line $\overleftrightarrow{AB}\&\overleftrightarrow{A\text{'}B\text{'}}$ are perpendicular.

Hence,

Both line $\overleftrightarrow{AB}\&\overleftrightarrow{A\text{'}B\text{'}}$ are perpendicular.

d.

To determine

To proof: Theorem $AB=A\text{'}B\text{'}$

Answer to Problem 33WE

  $AB=A\text{'}B\text{'}$ since translation in as isometry.

Explanation of Solution

Given:

  $A(0,3)\&B(4,1)$ & their image $A\text{'}(3,0)\&B\text{'}(1,-4)$ .

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.

Calculation:

As per the given problem

  $AB=A\text{'}B\text{'}$

Since translation in as isometry.

For reference

  

Hence,

  $AB=A\text{'}B\text{'}$

e.

To determine

To compare: Area of $\Delta AOB\&\Delta A\text{'}OB\text{'}$

Answer to Problem 33WE

Area of $\Delta AOB\&\Delta A\text{'}OB\text{'}$ are same.

Explanation of Solution

Given:

  $A(0,3)\&B(4,1)$ & their image $A\text{'}(3,0)\&B\text{'}(1,-4)$ .

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.

Calculation:

As per the given problem

  $A(0,3)\&B(4,1)$ & their image $A\text{'}(3,0)\&B\text{'}(1,-4)$ .

  

We know rotation preserves distance i.e. invariant.

  $\begin{array}{l}OA=OA\text{'}\\ OB=OB\text{'}\\ AB=A\text{'}B\text{'}\end{array}$

Since all sides of triangle are of same length, then area will be same.

Hence,

Area of $\Delta AOB\&\Delta A\text{'}OB\text{'}$ are same.

f.

To determine

To find: Image of $(x,y)$ by $R_{O,270}$

Answer to Problem 33WE

  $(x,y)\to (y,-x)$ by $R_{O,270}$ .

Explanation of Solution

Given:

Coordinate $A(0,3)\&B(4,1)$ .

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.

Calculation:

As per the given problem

  $A(0,3)\&B(4,1)$ & their image $A\text{'}(3,0)\&B\text{'}(1,-4)$ .

  

Hence,

  $(x,y)\to (y,-x)$ by $R_{O,270}$ .