Chapter 7.6, Problem 27HP

To determine

Explain the enlargement, reduction, or a congruence transformationby the help of scale factor.

Explanation of Solution

Given:

The given statement is that the enlargement transformation.

Calculation:

A dilation with a scale factor greater than one produces an enlargement, or an image that is larger than the original figure.

If $K>1$ , the dilation is an enlargement.

Where $K=$ scale factor.

Example

  

The dilation is an enlargement because $B$ is larger than $A$ .

From the diagram, for the original figure $A$ .

The distance between the vertices at $\left(0,0\right)$ and $\left(4,0\right)$  $=4$

For the image figure $B$ .

The distance between the vertices at $\left(0,0\right)$ and $\left(8,0\right)$  $=8$

Scale factor $=\frac{\text{dimension}\text{of}\text{image}}{\text{dimension}\text{of}\text{original}}=\frac{8}{4}=2$

Hence the dilation is an enlargement and Scale factor is $2$ .

A dilation with a scale factor between zero and one produces a reduction, an image that is smaller than the original figure.

If $0<K>1$ , the dilation is a reduction.

Where $K=$ scale factor.

Example

  

The dilation is a reduction because $B$ is smaller than $A$ .

From the diagram, for the original figure $A$ .

The distance between the vertices at $\left(-3,2\right)$ and $\left(3,2\right)$  $=6$

For the image figure $B$ .

The distance between the vertices at $\left(-1.5,1\right)$ and $\left(15,1\right)$  $=3$

Scale factor $=\frac{\text{dimension}\text{of}\text{image}}{\text{dimension}\text{of}\text{original}}=\frac{3}{6}=\frac{1}{2}$

Hence the dilation is a reduction and Scale factor is $\frac{1}{2}$ .

Two objects are congruent if they have the same size and shape. In a congruence transformation, an object transformed into a congruent object. Scale factor of the congruence transformationis $K=1$ , Where $K=$ scale factor.

Example

  

From the diagram, for the original figure $WXYZ$ .

The distance between the vertices at $\left(0,0\right)$ and $\left(0,2\right)$  $=2$

For the image figure $ABCD$ .

The distance between the vertices at $\left(0,0\right)$ and $\left(0,2\right)$  $=2$

Scale factor $=\frac{\text{dimension}\text{of}\text{image}}{\text{dimension}\text{of}\text{original}}=\frac{2}{2}=1$

Hence the dilation is a congruence transformation and Scale factor is $1$ .