Chapter 6.8, Problem 23HP

To determine

To find: The pair of points that does not represents dilation with center at origin.

Answer to Problem 23HP

Pair $\text{C}\left(2,5\right)\to \text{C`}\left(4,7\right)$does not follow the said dilation as the y coordinate of imaged coordinate C ` are 7 that should be 10.

Explanation of Solution

Given information : 

Following pair of points are given:

  

Concept used:

By rule, in dilation with center at origin, each original coordinate $(x,y)$ is changed as:

  $\left(x,y\right)\to (cx,cy)$

Where, c is the scale factor of dilation.

Calculation: Now, in first pair,

  $\text{A}\left(2,3\right)\to \text{A`}(2\times 2,2\times 3)$

Or, applied scale factor is 2.

Similarly, in second pair,

  $\text{B}\left(4,6\right)\to \text{B`}\left(\frac{1}{2}\times 4,\frac{1}{2}\times 6\right)=\text{B`}(2,3)$

or, applied scale factor is $\frac{1}{2}$ .

In third pair,

  $\text{C}\left(2,5\right)\to \text{C`}\left(2\times 2,2\times 5\right)=\text{C`}(4,10)$

So, scale factor to be applied here is 2.

And in four pair of points,

  $\text{D}\left(-2,4\right)\to \text{D`}\left(-\frac{1}{2}\times 2,\frac{1}{2}\times 4\right)=\text{D`}(-1,2)$

Or applied scale factor is $\frac{1}{2}$ .

Above calculation, it is cleared that in third pair of points, the imaged point C` is not following the proper dilation with scale factor 2. So, the pair $\text{C}\left(2,5\right)\to \text{C`}\left(4,7\right)$ does not belong to said dilation.