# A line segment has endpoints at (1 ,2 ) and (3 , 1). The line segment is dilated by a factor of 4  around (2 , 5). What are the new endpoints and length of the line segment?

Feb 5, 2018

New end points color(brown)((-2, -7) & color(brown)((6, -11)

Length of the line segment $\textcolor{g r e e n}{d \approx 8.94}$

#### Explanation:

Given : End points A(1,2), B (3,1), Center of dilation C(2,5) and dilation factor 4

Let A' and B' be the new end points after dilation.

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

$a ' - c = 4 \cdot \left(a - c\right)$

$a ' = 4 a - 3 c$

$\implies 4 \left(\begin{matrix}1 \\ 2\end{matrix}\right) - 3 \left(\begin{matrix}2 \\ 5\end{matrix}\right)$

$\implies \left(\begin{matrix}4 \\ 8\end{matrix}\right) - \left(\begin{matrix}6 \\ 15\end{matrix}\right) = \left(\begin{matrix}- 2 \\ - 7\end{matrix}\right)$

color(brown)(A' (-2, -7)

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

$b ' - c = 4 \left(b - c\right)$

$b ' = 4 b - 3 c$

$\implies 4 \left(\begin{matrix}3 \\ 1\end{matrix}\right) - 3 \left(\begin{matrix}2 \\ 5\end{matrix}\right)$

$\implies \left(\begin{matrix}12 \\ 4\end{matrix}\right) - \left(\begin{matrix}6 \\ 15\end{matrix}\right) = \left(\begin{matrix}6 \\ - 11\end{matrix}\right)$

color(brown)(B' (6, -11)

Using distance formula we can find the length A'B'

bar(A'B') = sqrt((6-(-2))^2 + ((-11) - (-7))^2) = sqrt(8^2 + 4^2) ~~ color(green)(8.94 corrected to two decimal points