# A line segment has endpoints at (2 ,4 ) and (5 ,3 ). The line segment is dilated by a factor of 3  around (3 ,8 ). What are the new endpoints and length of the line segment?

Dec 29, 2017

New end points: $\hat{P} : \left(0 , - 4\right)$ and $\hat{Q} : \left(12 , 5\right)$
New segment length: $\left\mid \hat{P} \hat{Q} \right\mid = 15$

#### Explanation:

Let $P$ be the original point $\left(2 , 4\right)$
Let $Q$ be the original point $\left(5 , 3\right)$
and
Let $C$ be the center of dilation, $\left(3 , 8\right)$

Consider the vector $\vec{C P}$
$\textcolor{w h i t e}{\text{XXX}} \vec{C P} = \left(2 , 4\right) - \left(3 , 8\right) = \left(- 1 , - 4\right)$
Dilation by a factor of $3$ will scale this vector up by a factor of $3$
So $P$ will move to the new location:
$\textcolor{w h i t e}{\text{XXX}} \hat{P} = C + 3 \vec{C P}$
$\textcolor{w h i t e}{\text{XXXX}} = \left(3 , 8\right) + 3 \left(- 1 , 4\right)$
$\textcolor{w h i t e}{\text{XXXX}} = \left(3 - 3 , 8 - 12\right)$
$\textcolor{w h i t e}{\text{XXXX}} = \left(0 , - 4\right)$

Similarly
$\textcolor{w h i t e}{\text{XXX}} \vec{C Q} = \left(5 , 3\right) - \left(2 , 4\right) = \left(3 , - 1\right)$
and new location for $Q$ at
$\textcolor{w h i t e}{\text{XXX}} \hat{Q} = \left(3 , 8\right) + 3 \left(3 , - 1\right)$
$\textcolor{w h i t e}{\text{XXXX}} = \left(12 , 5\right)$

The length of the new line segment will be (using the Pythagorean Theorem)
$\textcolor{w h i t e}{\text{XXX}} \left\mid \hat{P} \hat{Q} \right\mid = \sqrt{{\left(12 - 0\right)}^{2} + {\left(5 - \left(- 4\right)\right)}^{2}}$
$\textcolor{w h i t e}{\text{XXX}} = \sqrt{{12}^{2} + {9}^{2}}$
$\textcolor{w h i t e}{\text{XXX}} = \sqrt{225}$
$\textcolor{w h i t e}{\text{XXX}} = 15$