# A line segment has endpoints at (1 ,4 ) and (3 ,4 ). The line segment is dilated by a factor of 6  around (2 ,5 ). What are the new endpoints and length of the line segment?

Oct 17, 2017

New endpoints: $\left(- 4 , - 1\right) , \left(8 , - 1\right)$
New line segment length: $12$

#### Explanation:

Note that the distance between the initial line segment end points: $\left(1 , 4\right)$ and $\left(3 , 4\right)$ is $\textcolor{g r e e n}{2}$ units.
Dilating by a factor of $\textcolor{m a \ge n t a}{6}$ will result in a new length of $\textcolor{m a \ge n t a}{6} \times \textcolor{g r e e n}{2} = \textcolor{red}{12}$

Consider the vectors from the point of dilation: $\left(2 , 5\right)$ and each of the end points:
{: ("Initial endpoints:",color(white)("xxx"),(1,4),color(white)("xxx"),(3,4)), ("vector from dilation point "(2,5)":",color(white)("xxx"),(-1,-1),color(white)("xxx"),(1,-1)), ("vector after dilation " 6xx":"color(white)("xxx"),,(-6,-6),color(white)("xxx"),(6,-6)), ("new end points:",color(white)("xxx"),(2,5)+(-6,-6),color(white)("xxx"),(2,5)+(6,-6)), (,color(white)("xxx"),=color(red)(""(-4,-1)),color(white)("xxx"),=color(red)(""(8,-1))) :}