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

Jun 28, 2017

The new endpoints are $\left(4 , \frac{7}{2}\right)$ and $\left(3 , \frac{7}{2}\right)$ and the length of the line segment is $= 1$

#### Explanation:

Let the line segment be $A B$

$A = \left(5 , 5\right)$ and

$B = \left(3 , 5\right)$

$C = \left(3 , 2\right)$

The length of the segment $A B = 2$

Let point $A ' = \left(x , y\right)$ be the point after dilatation, then

$\vec{C A '} = \frac{1}{2} \vec{C A}$

$\left(\begin{matrix}x - 3 \\ y - 2\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}5 - 3 \\ 5 - 2\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}2 \\ 3\end{matrix}\right)$

Therefore,

$x - 3 = 1$, $\implies$, $x = 4$

$y - 2 = \frac{3}{2}$, $\implies$, $y = \frac{7}{2}$

The point $A ' = \left(4 , \frac{7}{2}\right)$

Let point $B ' = \left(p , q\right)$ be the point after dilatation, then

$\vec{C B '} = \frac{1}{2} \vec{C B}$

$\left(\begin{matrix}p - 3 \\ q - 2\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}3 - 3 \\ 5 - 2\end{matrix}\right) = \frac{1}{2} \left(\begin{matrix}0 \\ 3\end{matrix}\right)$

Therefore,

$p - 3 = 0$, $\implies$, $p = 3$

$q - 2 = \frac{3}{2}$, $\implies$, $q = \frac{7}{2}$

The point $B ' = \left(3 , \frac{7}{2}\right)$

The length of the segment $A ' B ' = 1$