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

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After dilating the line segment about the point (2, 2) by a factor of 3, the new endpoints are (9, 5) and (0, 11).

After reflecting across the line x = 4, the new endpoints become (5, 5) and (8, 11).

Then, reflecting across the line y = -1, the new endpoints are (5, 11) and (8, 5).

The distance of these endpoints from the origin can be calculated using the distance formula ( \sqrt{(x - x_0)^2 + (y - y_0)^2} ), where ( (x_0, y_0) ) is the origin (0, 0).

For the endpoint (5, 11): ( \sqrt{(5 - 0)^2 + (11 - 0)^2} = \sqrt{25 + 121} = \sqrt{146} )

For the endpoint (8, 5): ( \sqrt{(8 - 0)^2 + (5 - 0)^2} = \sqrt{64 + 25} = \sqrt{89} )

So, the distances from the new endpoints to the origin are approximately ( \sqrt{146} ) and ( \sqrt{89} ) units.

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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.

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