Chapter 6, Problem 44SGR

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 44SGR

Vertices of figure after dilation are $L^{\prime }\left(-1,-\frac{1}{2}\right),M^{\prime }\left(-\frac{1}{2},1\right),N^{\prime }\left(1,0\right)$

Explanation of Solution

Given information: Vertices of figure is $L\left(-4,-2\right),M\left(-2,4\right),N\left(4,0\right),k=\frac{1}{4}$

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:

  $L\left(-4,-2\right),M\left(-2,4\right),N\left(4,0\right)$

Also, scale factor is

  $k=\frac{1}{4}$

Vertices of figure after dilation are calculated as follows:

  $\begin{array}{l}L\left(-4,-2\right)\to L^{\prime }\left(-4\times \frac{1}{4},-2\times \frac{1}{4}\right)\\ M\left(-2,4\right)\to M^{\prime }\left(-2\times \frac{1}{4},4\times \frac{1}{4}\right)\\ N\left(4,0\right)\to N^{\prime }\left(4\times \frac{1}{4},0\times \frac{1}{4}\right)\end{array}$

On solving,

  $\begin{array}{l}L\left(-4,-2\right)\to L^{\prime }\left(-1,-\frac{1}{2}\right)\\ M\left(-2,4\right)\to M^{\prime }\left(-\frac{1}{2},1\right)\\ N\left(4,0\right)\to N^{\prime }\left(1,0\right)\end{array}$

Hence, vertices of figure after dilation are $L^{\prime }\left(-1,-\frac{1}{2}\right),M^{\prime }\left(-\frac{1}{2},1\right),N^{\prime }\left(1,0\right)$

Now, on graphing the original image and dilation image