Chapter 6, Problem 43SGR

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 43SGR

Vertices of figure after dilation are $Q^{\prime }\left(-6,9\right),R^{\prime }\left(3,6\right),S^{\prime }\left(9,-3\right),T^{\prime }\left(-6,-6\right)$

Explanation of Solution

Given information: Vertices of figure is $Q\left(-2,3\right),R\left(1,2\right),S\left(3,-1\right),T\left(-2,-2\right),k=3$

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:

  $Q\left(-2,3\right),R\left(1,2\right),S\left(3,-1\right),T\left(-2,-2\right)$

Also, scale factor is

  $k=3$

Vertices of figure after dilation are calculated as follows:

  $\begin{array}{l}Q\left(-2,3\right)\to Q^{\prime }\left(-2\times 3,3\times 3\right)\\ R\left(1,2\right)\to R^{\prime }\left(1\times 3,2\times 3\right)\\ S\left(3,-1\right)\to S^{\prime }\left(3\times 3,-1\times 3\right)\\ T\left(-2,-2\right)\to T^{\prime }\left(-2\times 3,-2\times 3\right)\end{array}$

On solving,

  $\begin{array}{l}Q\left(-2,3\right)\to Q^{\prime }\left(-6,9\right)\\ R\left(1,2\right)\to R^{\prime }\left(3,6\right)\\ S\left(3,-1\right)\to S^{\prime }\left(9,-3\right)\\ T\left(-2,-2\right)\to T^{\prime }\left(-6,-6\right)\end{array}$

Hence, vertices of figure after dilation are $Q^{\prime }\left(-6,9\right),R^{\prime }\left(3,6\right),S^{\prime }\left(9,-3\right),T^{\prime }\left(-6,-6\right)$

Now, on graphing the original image and dilation image