Chapter 6, Problem 15PT

To determine

To calculate:Tofind the vertices of triangle $D^{\prime }{E}^{\prime }{F}^{\prime }$ after a dilation of $\frac{1}{2}$

Answer to Problem 15PT

Vertices of triangle $D^{\prime }{E}^{\prime }{F}^{\prime }$ are $D^{\prime }\left(2,8\right),E^{\prime }\left(6,-6\right),F^{\prime }\left(2,-4\right)$

Explanation of Solution

Given information: Triangle $DEF$ has vertices $D\left(4,16\right),E\left(12,-12\right),F\left(4,-8\right)$

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:

On plotting triangle $DEF$ on graph,

Given scaling factor, $m=\frac{1}{2}$

According to dilation,

Dilation with scale factor $m$ means

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

Thus,

$\begin{array}{l}D\left(4,16\right)\to D^{\prime }\left(4\times \frac{1}{2},16\times \frac{1}{2}\right)\\ E\left(12,-12\right)\to E^{\prime }\left(12\times \frac{1}{2},-12\times \frac{1}{2}\right)\\ F\left(4,-8\right)\to F^{\prime }\left(4\times \frac{1}{2},-8\times \frac{1}{2}\right)\end{array}$

On solving,

$\begin{array}{l}D\left(4,16\right)\to D^{\prime }\left(2,8\right)\\ E\left(12,-12\right)\to E^{\prime }\left(6,-6\right)\\ F\left(4,-8\right)\to F^{\prime }\left(2,-4\right)\end{array}$

Thus, vertices of triangle $D^{\prime }{E}^{\prime }{F}^{\prime }$ are $D^{\prime }\left(2,8\right),E^{\prime }\left(6,-6\right),F^{\prime }\left(2,-4\right)$

Hence, correct option is H