Proof that Isometry f Preserves Midpoints | Geometric Reflection Counterexample

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In summary, the conversation discusses proving that an isometry that fixes the origin also preserves midpoints of line segments. There is some initial confusion about the possibility of a reflection not preserving midpoints, but it is clarified that a reflection still transforms midpoints into midpoints. The challenge is then to prove this algebraically, using the fact that an isometry preserves vector norms and distances. The conversation also mentions the need to show that f(ru)=rf(u) under the given conditions.
  • #1
pivoxa15
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Homework Statement


Suppose f is an isometry that fixes O (origin). Prove f preserves midpoints of line segments.


The Attempt at a Solution


Geometricallly, f could be a reflection in which case it would not preserve the mid point of any line segment that does not intersect the origin anywhere.

So I don't see a proof at all and infact sees a mistake.
 
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  • #2
But in the case of a reflection the transformation of a midpoint is still a midpoint, no?
 
  • #3
That is true. I was thinking along the wrong lines (no pun intended) in that I was thinking that f maps midpoint to the exact same mid point.

Everything makes geometric sense. The only problem is to prove it algebraically. Can't see how to do it.
 
  • #4
Are we in R^n?
 
  • #5
If f is a isometry, u.v=f(u).f(v) holds. So the vector norm (u.u)^1/2 and distance stays the same.
 
Last edited:
  • #6
I have worked out the quesion in the OP. I now need to show that f(ru)=rf(u) with the same conditions given in the OP.
 

1. What does "F preserves midpoint" mean?

"F preserves midpoint" means that given two points A and B, the midpoint of the line segment AB remains the same after applying the function F to both points.

2. How can I determine if a function preserves midpoint?

To determine if a function preserves midpoint, you can plug in the coordinates of two points A and B into the function and see if the resulting midpoint is the same as the original midpoint of the line segment AB.

3. What are some common examples of functions that preserve midpoint?

Some common examples of functions that preserve midpoint include translations, reflections, rotations, and dilations.

4. How is the preservation of midpoint important in mathematics?

The preservation of midpoint is important in mathematics because it is a property that is often used in geometric proofs and constructions. It also helps us understand the behavior of functions and their effects on geometric figures.

5. Can a function preserve midpoint but not be an isometry?

Yes, a function can preserve midpoint but not be an isometry. This means that the function may preserve the length and midpoint of a line segment, but it does not necessarily preserve the distance between all points in the figure.

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