# A line segment has endpoints at (9 ,7 ) and (5 ,4 ). The line segment is dilated by a factor of 3  around (4 ,3 ). What are the new endpoints and length of the line segment?

Feb 10, 2018

New end points $A ' \left(\begin{matrix}5 \\ 4\end{matrix}\right) , B ' \left(\begin{matrix}1 \\ 1\end{matrix}\right)$

Length of line segment $\vec{A ' B '} = \textcolor{g r e e n}{5}$

#### Explanation:

Given $A \left(\begin{matrix}9 \\ 7\end{matrix}\right) , B \left(\begin{matrix}5 \\ 4\end{matrix}\right) , C \left(\begin{matrix}4 \\ 3\end{matrix}\right)$ & dilation factor 3

$\vec{A ' C} = 4 \cdot \vec{A C}$

$A ' \left(\begin{matrix}x - 4 \\ y - 3\end{matrix}\right) = 3 \cdot \left(\begin{matrix}9 - 4 \\ 7 - 3\end{matrix}\right) \implies \left(\begin{matrix}5 \\ 4\end{matrix}\right)$

$A ' \left(\begin{matrix}9 \\ 7\end{matrix}\right)$

$\vec{B ' C} = 4 \cdot \vec{B C}$

B' ((x-4),(y-3)) = 3 * ((5-4),4-3))=> ((1),(1))

$B ' \left(\begin{matrix}5 \\ 4\end{matrix}\right)$

Length of line segment $\vec{A ' B '} = \sqrt{{\left(5 - 1\right)}^{2} + {\left(4 - 1\right)}^{2}} = \textcolor{g r e e n}{5}$