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

New endpoints $\left(\frac{7}{4} , \frac{17}{4}\right)$ and $\left(\frac{9}{4} , 4\right)$
New Length $l = \frac{\sqrt{5}}{4}$

#### Explanation:

First Endpoint $\left({x}_{1} , {y}_{1}\right)$
For ${x}_{1} :$

$\frac{2 - {x}_{1}}{2 - 1} = \frac{1}{4}$

${x}_{1} = \frac{7}{4}$

For ${y}_{1} :$

$\frac{5 - {y}_{1}}{5 - 2} = \frac{1}{4}$

${y}_{1} = \frac{17}{4}$

$\left(\frac{7}{4} , \frac{17}{4}\right)$

Second Endpoint $\left({x}_{2} , {y}_{2}\right)$

For ${x}_{2} :$

$\frac{2 - {x}_{2}}{2 - 3} = \frac{1}{4}$

${x}_{2} = \frac{9}{4}$

For ${y}_{2} :$

$\frac{5 - {y}_{2}}{5 - 1} = \frac{1}{4}$

${y}_{2} = 4$

$\left(\frac{9}{4} , 4\right)$

Length $l = \sqrt{{\left({x}_{2} - {x}_{1}\right)}^{2} + {\left({y}_{2} - {y}_{1}\right)}^{2}}$

$l = \sqrt{{\left(\frac{9}{4} - \frac{7}{4}\right)}^{2} + {\left(4 - \frac{17}{4}\right)}^{2}}$

$l = \sqrt{{\left(\frac{2}{4}\right)}^{2} + {\left(- \frac{1}{4}\right)}^{2}}$

$l = \frac{\sqrt{5}}{4}$

God bless... I hope the explanation is useful