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Question 1161626: Prove that, given a line and its midpoint, the transformed midpoint will still be the midpoint of the transformed line.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
The assertion made in your post cannot be proven. A line, being of infinite length, cannot have a midpoint -- it has no endpoints. However, presuming you meant Prove that, given a line segment, and its midpoint, the transformed midpoint will still be the midpoint of the transformed line., the following is a valid proof.
Let ) and ) be the endpoints of an arbitrary line segment. Suppose the given line segment is translated  units in the horizontal direction and  units in the vertical. Additionally, suppose the given line segment is dilated by a factor of  . Hence the transformed line segment would have endpoints of ,r(y_1\,+\,k))) and ,r(y_2\,+\,k)))
The  -coordinate of the midpoint of the original line is given by:
We need to show that the expression for the transformed midpoint coordinate, ) , is identical to the expression for the transformed midpoint in terms of the -coordinates of the transformed endpoints.
\ +\ r(x_2\,+\,h)}{2})
\ +\ rh)
)
The proof for the  -coordinate of the midpoint is essentially identical.
QED
John
My calculator said it, I believe it, that settles it
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