# A line segment goes from (3 ,4 ) to (5 ,1 ). The line segment is dilated about (1 ,0 ) by a factor of 2. Then the line segment is reflected across the lines x=-2 and y=2, in that order. How far are the new endpoints from the origin?

Apr 21, 2018

If I've done it right the first is $\sqrt{85}$ from the origin and the second is $11$ from the origin.

#### Explanation:

Thought I just did one of these.

Shifting the dilation point to the origin maps the other points:

$\left(3 , 4\right) - \left(1 , 0\right) = \left(2 , 4\right) \quad \quad \quad \quad \left(5 , 1\right) - \left(1 , 0\right) = \left(4 , 1\right)$

Doubling each dilates around the origin by a factor of two.

$2 \left(2 , 4\right) = \left(4 , 8\right) \quad \quad \quad \quad 2 \left(4 , 1\right) = \left(8 , 2\right)$

Now we shift the origin back to the dilation point:

$\left(4 , 8\right) + \left(1 , 0\right) = \left(5 , 8\right) \quad \quad \quad \quad \left(8 , 2\right) + \left(1 , 0\right) = \left(9 , 2\right)$

Now we reflect through $x = - 2$ which leaves the $y$ coordinate alone.

$\left(- 2 - 5 , 8\right) = \left(- 7 , 8\right) \quad \quad \quad \quad \left(- 2 - 9 , 2\right) = \left(- 11 , 2\right)$

Now we reflect through $y = 2$ which leaves the $x$ coordinate alone.

$\left(- 7 , 2 - 8\right) = \left(- 7 , - 6\right) \quad \quad \quad \quad \left(- 11 , 2 - 2\right) = \left(- 11 , 0\right)$

If I've done that right the first is $\sqrt{{7}^{2} + {6}^{2}} = \sqrt{85}$ from the origin and the second is $11$ from the origin.