# A line segment has endpoints at (3 ,2 ) and (7 , 5). The line segment is dilated by a factor of 4  around (2 , 3). What are the new endpoints and length of the line segment?

Apr 21, 2017

$\left(6 , - 1\right) , \left(22 , 11\right) , 20 \text{ units}$

#### Explanation:

let $A = \left(3 , 2\right) , B = \left(7 , 5\right) \text{ and } C = \left(2 , 3\right)$

$\text{and " A',B'" be the image of A and B under the dilatation}$

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

$\Rightarrow \vec{C A '} = 4 \left(\begin{matrix}1 \\ - 1\end{matrix}\right) = \left(\begin{matrix}4 \\ - 4\end{matrix}\right)$

$\Rightarrow A ' = \left(2 + 4 , 3 - 4\right) = \left(6 , - 1\right) \textcolor{red}{\leftarrow}$

$\vec{C B} = \underline{b} - \underline{c} = \left(\begin{matrix}7 \\ 5\end{matrix}\right) - \left(\begin{matrix}2 \\ 3\end{matrix}\right) = \left(\begin{matrix}5 \\ 2\end{matrix}\right)$

$\Rightarrow \vec{C B '} = 4 \left(\begin{matrix}5 \\ 2\end{matrix}\right) = \left(\begin{matrix}20 \\ 8\end{matrix}\right)$

$\Rightarrow B ' = \left(2 + 20 , 3 + 8\right) = \left(22 , 11\right) \textcolor{red}{\leftarrow}$

To calculate length use the $\textcolor{b l u e}{\text{distance formula}}$

$\textcolor{red}{\overline{\underline{| \textcolor{w h i t e}{\frac{2}{2}} \textcolor{b l a c k}{d = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}} \textcolor{w h i t e}{\frac{2}{2}} |}}}$

$\text{let " (x_1,y_1)=(6,-1)" and } \left({x}_{2} , {y}_{2}\right) = \left(22 , 11\right)$

${d}_{A ' B '} = \sqrt{{\left(22 - 6\right)}^{2} + {\left(11 + 1\right)}^{2}}$

$\textcolor{w h i t e}{{d}_{A ' B '}} = \sqrt{256 + 144} = \sqrt{400}$

$\Rightarrow {d}_{A ' B '} = 20 \text{ units}$