# A line segment goes from (4 ,1 ) to (2 ,3 ). The line segment is dilated about (2 ,2 ) by a factor of 2. Then the line segment is reflected across the lines x = -2 and y=4, in that order. How far are the new endpoints form the origin?

Apr 21, 2018

I get $4 \sqrt{2}$ for the first endpoint and $0$ for the second.

#### Explanation:

Yikes, that's a lot of transformations. We're just interested in the image of each endpoint.

To do the dilation we start by getting a direction vector from the dilation point to each endpoint, essentially translating the dilation point to the origin.

$\left(4 , 1\right) - \left(2 , 2\right) = \left(2 , - 1\right) \quad \quad \quad \quad \left(2 , 3\right) - \left(2 , 2\right) = \left(0 , 1\right)$

We dilate each direction vector by a factor of two and translate back:

$\left(2 , 2\right) + 2 \left(2 , - 1\right) = \left(6 , 0\right) \quad \quad \quad \quad \left(2 , 2\right) + 2 \left(0 , 1\right) = \left(2 , 4\right)$

Reflecting through $x = - 2$ leaves the $y$ coordinate alone:

$\left(6 , 0\right) \to \left(2 - 6 , 0\right) = \left(- 4 , 0\right) \quad \quad \quad \quad \left(2 , 4\right) \to \left(2 - 2 , 4\right) = \left(0 , 4\right)$

Reflecting through $y = 4$ leaves the $x$ coordinate alone:

$\left(- 4 , 0\right) \to \left(- 4 , 4 - 0\right) = \left(- 4 , 4\right) \quad \quad \quad \quad \left(0 , 4\right) \to \left(0 , 0\right)$

If I did that right the first endpoint is $\setminus \sqrt{{4}^{2} + {4}^{2}} = 4 \setminus \sqrt{2}$ from the origin and the second endpoint is the origin, so a distance of zero.