# A line segment has endpoints at (1 ,4 ) and (4 , 9). The line segment is dilated by a factor of 1/3  around (4 , 2). What are the new endpoints and length of the line segment?

Jun 7, 2018

color(brown)("Length of the line segment " color(blue)(bar(AB) " after dilation is " color(green)(1.94 " units"

#### Explanation:

$A \left(1 , 4\right) , B \left(4 , 9\right) . \text{ Dilated by a factor of 1/3 around } C \left(4 , 2\right)$

$\overline{A ' C} = \left(\frac{1}{3}\right) \overline{A C}$

$A ' C - c = \left(\frac{1}{3}\right) \left(a - c\right)$

$A ' = \frac{a}{3} + \left(\frac{2}{3}\right) c$

$A ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{3}\right) \left(\begin{matrix}1 \\ 4\end{matrix}\right) + \left(\frac{2}{3}\right) \left(\begin{matrix}4 \\ 2\end{matrix}\right) = \left(\begin{matrix}3 \\ \frac{8}{3}\end{matrix}\right)$

Similarly, $B ' = \frac{b}{3} + \left(\frac{2}{3}\right) c$

$B ' \left(\begin{matrix}x \\ y\end{matrix}\right) = \left(\frac{1}{3}\right) \left(\begin{matrix}4 \\ 9\end{matrix}\right) + \left(\frac{2}{3}\right) \left(\begin{matrix}4 \\ 2\end{matrix}\right) = \left(\begin{matrix}4 \\ \frac{13}{3}\end{matrix}\right)$

Length of the line segment $A ' B ' = \sqrt{{\left(3 - 4\right)}^{2} + {\left(\frac{8}{3} - \frac{13}{3}\right)}^{2}}$

vec(A'B') = 1.94 ' units"