# A line segment has endpoints at (5 ,9 ) and (6 ,7 ). The line segment is dilated by a factor of 4  around (5 ,1 ). What are the new endpoints and length of the line segment?

Mar 9, 2017

$\left(5 , 33\right)$ & $\left(9 , 24\right)$

#### Explanation:

Start by drawing the line in a rectangular coordinate plane and locating the point of dilation $\left(5 , 1\right)$.

Draw lines from $\left(5 , 1\right)$ through each endpoint, extending above.

Since the point $\left(5 , 1\right)$ and $\left(5 , 9\right)$ have a $y$-difference of $8$. Multiply this by $4$ to get $32$. Add $1$ to get to the point location from $\left(5 , 1\right)$.

The slope of the original line = $- \frac{2}{1}$. From the point $\left(5 , 33\right)$, apply this slope $4$ times to find the second endpoint. It needs to lie on the line from $\left(5 , 1\right)$ through $\left(6 , 7\right)$.

You can see the process below:

The dilated line segment should be $4$ times bigger.

Original line segment length $= \sqrt{{1}^{2} + {2}^{2}} = \sqrt{5} \approx 2.236$

Dilated line segment length = $\sqrt{{8}^{2} + {4}^{2}} = \sqrt{80} = \sqrt{16 \cdot 5} = 4 \sqrt{5} \approx 8.9443$

So endpoints of the dilated line are: $\left(5 , 33\right)$ & $\left(9 , 24\right)$