Geometry/Topology Isometry question

[kimberu]

Homework Statement

Let P, Q, P' and Q' be 4 points in the euclidean plane where distance(P,Q) = distance(P', Q') and the distance is not 0. Show there are precisely 2 isometries $$\varphi$$ so $$\varphi(Q) = Q^{l}$$ and $$\varphi(P) = P^{l}$$

Homework Equations

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The Attempt at a Solution

So I know (I think) that there's at least one such isometry because the plane is isotropic, which means that a translation/rotation exists to map these points. But I don't know how to define 2 such $$\varphi$$s, or what the differences are between them.

Thank you so much for any help!

 

[mighty2000]

Thank you for your question. I would like to help you understand the concept of isometries in this scenario.

First, it is important to note that an isometry is a transformation that preserves distances between points. In other words, if we have two points P and Q, and we apply an isometry to both of them, the distance between the transformed points will remain the same as the distance between the original points.

Now, in this scenario, we have two sets of points - P and Q, and P' and Q' - that have the same distance between them. This means that there exists an isometry that can map P to P' and Q to Q', while preserving the distance between them. This is the isometry that you have mentioned in your attempt at a solution.

However, there is another type of isometry that can also satisfy this condition. This is the reflection isometry. A reflection isometry is a transformation that flips or mirrors an object across a line. In this scenario, we can reflect the points P and Q across a line to get P' and Q', while still preserving the distance between them.

Therefore, in total, there are two isometries that can map P to P' and Q to Q'. One is the translation/rotation isometry that you have mentioned, and the other is the reflection isometry. Both of these transformations will result in the same distance between the points, as required in the problem.

I hope this helps to clarify the concept of isometries in this scenario. Let me know if you have any further questions.