# A line segment has endpoints at (7 ,6 ) and (5 ,8 ). The line segment is dilated by a factor of 4  around (2 ,1 ). What are the new endpoints and length of the line segment?

Jan 30, 2017

$\left(22 , 21\right) , \left(14 , 29\right) \text{ and } 8 \sqrt{2}$

#### Explanation:

Let the endpoints be A (7 ,6) and B (5 ,8) and their images be A' and B', respectively, under the dilation.
Let the centre of dilatation be C (2 ,1)

$\vec{C A} = \underline{a} - \underline{c} = \left(\begin{matrix}7 \\ 6\end{matrix}\right) - \left(\begin{matrix}2 \\ 1\end{matrix}\right) = \left(\begin{matrix}5 \\ 5\end{matrix}\right)$

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

$\Rightarrow A ' = \left(2 + 20 , 1 + 20\right) = \left(\textcolor{red}{22 , 21}\right)$

Similar process to obtain coordinates of B'

$\vec{C B} = \underline{b} - \underline{c} = \left(\begin{matrix}5 \\ 8\end{matrix}\right) - \left(\begin{matrix}2 \\ 1\end{matrix}\right) = \left(\begin{matrix}3 \\ 7\end{matrix}\right)$

$\Rightarrow \vec{C B '} = 4 \left(\begin{matrix}3 \\ 7\end{matrix}\right) = \left(\begin{matrix}12 \\ 28\end{matrix}\right)$

$\Rightarrow B ' = \left(2 + 12 , 1 + 28\right) = \left(\textcolor{red}{14 , 29}\right)$

$\text{new endpoints are "(22,21)" and } \left(14 , 29\right)$

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

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

The 2 points here are (22 ,21) and (14 ,29)

let $\left({x}_{1} , {y}_{1}\right) = \left(22 , 21\right) \text{ and } \left({x}_{2} , {y}_{2}\right) = \left(14 , 29\right)$

$d = \sqrt{{\left(14 - 22\right)}^{2} + {\left(29 - 21\right)}^{2}} = \sqrt{64 + 64} = \sqrt{128}$

"length of line segment" =sqrt128=8sqrt2≈11.31