# A line segment has endpoints at (3 ,8 ) and (5 , 9). The line segment is dilated by a factor of 3  around (2 , 3). What are the new endpoints and length of the line segment?

May 23, 2018

color(blue)(A'=(5,18)
color(blue)(B'=(11,21)
$\textcolor{b l u e}{d = 3 \sqrt{5}}$

#### Explanation:

We can preform a dilation using vectors.

Let $A = \left(3 , 8\right)$, $B = \left(5 , 9\right)$, $D = \left(2 , 3\right)$ and $O$ be the origin.

$\vec{O D} = \left(\begin{matrix}2 \\ 3\end{matrix}\right)$

$\vec{D A} = \left(\begin{matrix}1 \\ 5\end{matrix}\right)$

$\vec{D B} = \left(\begin{matrix}3 \\ 6\end{matrix}\right)$

Dilating A:

$\vec{O A} = \vec{O D} + 3 \vec{D A} = \left(\begin{matrix}2 \\ 3\end{matrix}\right) + 3 \left(\begin{matrix}1 \\ 5\end{matrix}\right) = \left(\begin{matrix}5 \\ 18\end{matrix}\right)$

Dilating B:

$\vec{O B} = \vec{O D} + 3 \vec{D B} = \left(\begin{matrix}2 \\ 3\end{matrix}\right) + 3 \left(\begin{matrix}3 \\ 6\end{matrix}\right) = \left(\begin{matrix}11 \\ 21\end{matrix}\right)$

New endpoints:

$A ' = \left(5 , 18\right)$ and $B ' = \left(11 , 21\right)$

Length of line segment is found using the distance formula:

$d = \sqrt{{\left(11 - 5\right)}^{2} + {\left(21 - 18\right)}^{2}} = \sqrt{45} = \textcolor{b l u e}{3 \sqrt{5}}$