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

Dec 30, 2017

New endpoints: Dilation of $\left(2 , 1\right) \rightarrow \left(- 1 , 1\right)$
$\textcolor{w h i t e}{\text{New endpoints:}}$Dilation of $\left(4 , 2\right) \rightarrow \left(7 , 5\right)$
New line segment length: $4 \sqrt{5} \text{ units}$

#### Explanation:

If the center of dilation is $C = \left(3 , 1\right)$
and original endpoints are
{: (P=(2,1),color(white)("xxxxx"),Q=(4,2)), (vec(CP)=(2-3,1-1),,vec(CQ)=(4-3,2-1)), (color(white)(vec(CP))=(-1,0),,color(white)(vec(CQ))=(1,1)), ("dilated vector:",,"dilated vector:"), (4vec(CP)=(-4,0),,4vec(CQ)=(4,4)), (hat(P)=C+4vec(CP),,hat(Q)=C+4vec(CQ)), (color(white)(hat(P))=(-1,1),,color(white)(hat(Q))=(7,5)) :}

The length of the dilated line segment is
$\left\mid \hat{P} \hat{Q} \right\mid = \sqrt{{\left(7 - \left(- 1\right)\right)}^{2} + {\left(5 - 1\right)}^{2}} = \sqrt{80} = 4 \sqrt{5}$