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

##### 1 Answer

The new endpoints are

#### Explanation:

The most complicated part of this series of transformations is the first: dilating about a point that is *not* the origin. At first, this can seem daunting but it is much easier than expected. My method for doing this is as follows:

From here, You need treat the point of dilation, #(5,4) as the origin and use *relative distances* to your "origin".

To do this, you need to shift your point and the point of dilation so that the point of dilation is at the origin.

In this case, both points need to be moved down

Since we will do one axis of one point at a time,

Take the x-value of one of the points you want to dilate. In this case, we'll take the point

Subtract *relative*

Take the y-value of the same point that you took the x-value for. In this case, we'll take the

Subtract *relative*

You new point is

From here, apply your dilation of

However, you aren't done with dilation yet. We need to reverse the translation of the first point, because that isn't a translation we want to keep.

With your new point, add back the values that you previously subtracted. These values were

Re-adjusted, your first point with the complete dilation is

If you repeat this process with the other point you want to dilate (

In this picture, the purple line is your initial line segment. The orange *x* is your point of dilation, the red line is your dilated line segment, and the green lines show the path of dilation.

Modeled in equations,

Should give you your dilated points where

After that, the rest is easy. For reflecting over

The blue line is the final line segment.

You can use the Pythagorean Theorem (