A line segment has endpoints at (5 ,5 ) and (3 ,7 ). The line segment is dilated by a factor of 1/2  around (3 , 1). What are the new endpoints and length of the line segment?

Feb 5, 2018

New end points color(green)((4,3) and $\textcolor{g r e e n}{3 , 4}$

Length of the line segment after dilation is $\textcolor{p u r p \le}{1.4142}$

Explanation:

Given : A (5,5), B (3,7), Dilation Point c(3,1), Dilation factor 1/2

To find new end points A', B' and A'B'

$a ' - c = \left(\frac{1}{2}\right) \left(a - c\right)$

$a ' = \left(\frac{1}{2}\right) a + \left(\frac{1}{2}\right) c$

$a ' = \left(\frac{1}{2}\right) \left(\begin{matrix}5 \\ 5\end{matrix}\right) + \left(\frac{1}{2}\right) \left(\begin{matrix}3 \\ 1\end{matrix}\right) = \left(\begin{matrix}\frac{5}{2} \\ \frac{5}{2}\end{matrix}\right) + \left(\begin{matrix}\frac{3}{2} \\ \frac{1}{2}\end{matrix}\right) = \left(\begin{matrix}4 \\ 3\end{matrix}\right)$

$A ' \left(4 , 3\right)$

$b ' - c = \left(\frac{1}{2}\right) \left(b - c\right)$

$b ' = \left(\frac{1}{2}\right) b + \left(\frac{1}{2}\right) c = \left(\frac{1}{2}\right) \left(\begin{matrix}3 \\ 7\end{matrix}\right) + \left(\frac{1}{2}\right) \left(\begin{matrix}3 \\ 1\end{matrix}\right)$

$\implies \left(\begin{matrix}\frac{3}{2} \\ \frac{7}{2}\end{matrix}\right) + \left(\begin{matrix}\frac{3}{2} \\ \frac{1}{2}\end{matrix}\right) = \left(\begin{matrix}3 \\ 4\end{matrix}\right)$

$B ' \left(3 , 4\right)$

using distance formula between two points,

$\overline{A ' B '} = \sqrt{{\left(4 - 3\right)}^{2} + {\left(3 - 4\right)}^{2}} \approx 1.4142$