# 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 new endpoints are

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 (

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The new endpoints of the reflected line segment are approximately (4.8, -0.4) and (5.2, -7.6).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

- Point A is at #(1 ,-4 )# and point B is at #(-9 ,-8 )#. Point A is rotated #(3pi)/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?
- Circle A has a radius of #3 # and a center of #(8 ,7 )#. Circle B has a radius of #2 # and a center of #(6 ,1 )#. If circle B is translated by #<2 ,7 >#, does it overlap circle A? If not, what is the minimum distance between points on both circles?
- A line segment has endpoints at #(9 ,1 )# and #(5 ,3)#. If the line segment is rotated about the origin by #pi #, translated vertically by #-1 #, and reflected about the x-axis, what will the line segment's new endpoints be?
- Points A and B are at #(5 ,8 )# and #(8 ,1 )#, respectively. Point A is rotated counterclockwise about the origin by #pi # and dilated about point C by a factor of #5 #. If point A is now at point B, what are the coordinates of point C?
- Point A is at #(1 ,-4 )# and point B is at #(-9 ,-2 )#. Point A is rotated #(3pi)/2 # clockwise about the origin. What are the new coordinates of point A and by how much has the distance between points A and B changed?

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