# A line segment has endpoints at (4 ,3 ) and (3 ,9 ). The line segment is dilated by a factor of 4  around (2 ,2 ). What are the new endpoints and length of the line segment?

Jun 23, 2017

$\left(10 , 6\right) , \left(6 , 30\right) , \approx 24.33$

#### Explanation:

$\text{let " A(4,3),B(3,9),D(2,2)" and } A ' , B '$
$\text{be the image of A and B under the dilatation}$

$\vec{D A} = \underline{a} - \underline{d} = \left(\begin{matrix}4 \\ 3\end{matrix}\right) - \left(\begin{matrix}2 \\ 2\end{matrix}\right) = \left(\begin{matrix}2 \\ 1\end{matrix}\right)$

$\Rightarrow \vec{D A '} = \textcolor{red}{4} \vec{D A} = \textcolor{red}{4} \left(\begin{matrix}2 \\ 1\end{matrix}\right) = \left(\begin{matrix}8 \\ 4\end{matrix}\right)$

$\Rightarrow A ' = \left(2 + 8 , 2 + 4\right) = \left(10 , 6\right)$

$\vec{D B} = \underline{b} - \underline{d} = \left(\begin{matrix}3 \\ 9\end{matrix}\right) - \left(\begin{matrix}2 \\ 2\end{matrix}\right) = \left(\begin{matrix}1 \\ 7\end{matrix}\right)$

$\Rightarrow \vec{D B '} = \textcolor{red}{4} \vec{D B} = \textcolor{red}{4} \left(\begin{matrix}1 \\ 7\end{matrix}\right) = \left(\begin{matrix}4 \\ 28\end{matrix}\right)$

$\Rightarrow B ' = \left(2 + 4 , 2 + 28\right) = \left(6 , 30\right)$

$| A ' B ' | = \sqrt{{\left(6 - 10\right)}^{2} + {\left(30 - 6\right)}^{2}} = \sqrt{592}$

$\Rightarrow | A ' B ' | \approx 24.33 \text{ to 2 decimal places}$