# A line segment has endpoints at (3 ,2 ) and (2 , 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?

Feb 12, 2018

New coordinates of end points color(purple)(((6),(-1)), ((2),(11))

Length of the line segment vec(A'B') = ~~ color(green)(12.65

#### Explanation:

Existing points A (3,2), B (2,5). Dilated around C (2,3) by factor 4

To find new end points and length of segment.

$\vec{A ' C} = 4 \cdot \vec{A C} \mathmr{and} a ' = 4 a - 3 c$

$a ' = 4 \left(\begin{matrix}3 \\ 2\end{matrix}\right) - 3 \left(\begin{matrix}2 \\ 3\end{matrix}\right) = \left(\begin{matrix}12 \\ 8\end{matrix}\right) - \left(\begin{matrix}6 \\ 9\end{matrix}\right) \implies \left(\begin{matrix}6 \\ - 1\end{matrix}\right)$

$\vec{B ' C} = 4 \left(\vec{B C}\right) \mathmr{and} b ' = 4 a - 3 c$

$b ' = 4 \left(\begin{matrix}2 \\ 5\end{matrix}\right) - 3 \left(\begin{matrix}2 \\ 3\end{matrix}\right) = \left(\begin{matrix}8 \\ 20\end{matrix}\right) - \left(\begin{matrix}6 \\ 9\end{matrix}\right) = \left(\begin{matrix}2 \\ 11\end{matrix}\right)$

vec(A'B') = sqrt((2-6)^2 + (11+1)^) = sqrt160 ~~ color(green)(12.65