Chapter 6, Problem 45SGR

To determine

To calculate:Tofind the vertices of figure after a dilation with the given scale factor $k$ , $k$ centered about the origin. Also graph the original image and dilation image

Answer to Problem 45SGR

Vertices of figure after dilation are $F^{\prime }\left(2.5,7.5\right),G^{\prime }\left(7.5,10\right),H^{\prime }\left(5,2.5\right)$

Explanation of Solution

Given information: Vertices of figure is $F\left(1,3\right),G\left(3,4\right),H\left(2,1\right),k=2.5$

Formula Used: A dilation is a transformation that produces an image that is the same shape as the original, but it is different size.

Dilation with scale factor $m$ means

  $\left(x,y\right)\to \left(mx,my\right)$

Calculation:

Vertices of figure are as follows:

  $F\left(1,3\right),G\left(3,4\right),H\left(2,1\right)$

Also, scale factor is

  $k=2.5$

Vertices of figure after dilation are calculated as follows:

  $\begin{array}{l}F\left(1,3\right)\to F^{\prime }\left(1\times 2.5,3\times 2.5\right)\\ G\left(3,4\right)\to G^{\prime }\left(3\times 2.5,4\times 2.5\right)\\ H\left(2,1\right)\to H^{\prime }\left(2\times 2.5,1\times 2.5\right)\end{array}$

On solving,

  $\begin{array}{l}F\left(1,3\right)\to F^{\prime }\left(2.5,7.5\right)\\ G\left(3,4\right)\to G^{\prime }\left(7.5,10\right)\\ H\left(2,1\right)\to H^{\prime }\left(5,2.5\right)\end{array}$

Hence, vertices of figure after dilation are $F^{\prime }\left(2.5,7.5\right),G^{\prime }\left(7.5,10\right),H^{\prime }\left(5,2.5\right)$

Now, on graphing the original image and dilation image