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?

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 #4# and left #5#.

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 #(2,5) # whose x-value is #2#.

Subtract #5# from the #x#-value to get your relative #x#-value. You should get #-3#.

Take the y-value of the same point that you took the x-value for. In this case, we'll take the #5# from the same point, #(2,5)#.

Subtract #4# from the #y#-value to get your relative #y#-value. You should get #1#.

You new point is #(-3,1)#.

From here, apply your dilation of #3# to both point values and you should get a new point of #(-9,3)#.

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 #5# and #4#, for the #x# and #y# values respectively.

Re-adjusted, your first point with the complete dilation is #(-4,7)#.

If you repeat this process with the other point you want to dilate (#(3,2)#) you should come out with #(-1,-2)# as your final position.

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, #x_2=a(x_1-h)+h# and #y_2=a(y_1-k)+k#
Should give you your dilated points where #a# is your dilation factor, #h# is the #x#-value of your dilation point, and #k# is the #y#-value of your dilation point.

After that, the rest is easy. For reflecting over #x=4#, count the distance each point is from #x=4# and put it that many on the other side. Do the same for #y=-2#.

The blue line is the final line segment.

You can use the Pythagorean Theorem (#a^2+b^2=c^2#) to get the distances from the dilation point, where #a# is the #x# distance and #b# is the #y# distance.