Rigid Transformations and other topics -- help with Learning Geometry?

[cbarker1]

TL;DR Summary

I am asking this question because I am in the process of relearning high school geometry through Khan Academy. I am curious why Transformations, Dilations, and Symmetry help teaching geometry.

Dear Everybody,

I am in the process of relearning high school geometry through Khan Academy. I am current an graduated undergraduate student in mathematics. I am doing this because geometry is one of my weakest subject in mathematics. Second reason is that I want to reason out a problem geometrically. I also want to relearn my university level geometry textbook. I have a hard time with spatial reasoning in general. I am wondering why does learning the rigid transformations and dilations and symmetries help with learning high school geometry.Thanks,

cbarker1

 

[fresh_42]

I am curious why Transformations, Dilations, and Symmetry help teaching geometry.

You make a statement without giving any evidence or hints, and then ask why this is the case?

Why do you think this is of any help? Symmetries are equations and as such of help regardless in which area. Transformation and dilatation have little to do with geometry in its classical sense. All these jump in if we speak about analytical geometry, i.e. if we have a coordinate system. Geometry as taught at school (Pythagoras, Thales, triangle trigonometry, intercept theorem, etc.) doesn't need one, it only uses angles and lengths, and proportions.

 

[cbarker1]

The evidence is this list of topics in the khan academy. I have been working on the rigid transformation unit. I just want to know why is that case...

 

[suremarc]

A lot of arguments can be made by thinking about symmetry. Here’s a few arguments off the top of my head:

• An equilateral triangle (a triangle with equal sides) has equal angles due to its rotational and reflective symmetry.
• The area of a parallelogram is its base times its height, because it can be rearranged into a rectangle.
• The SAS, SSS, and ASA postulates can be proved by superimposing triangles on top of each other using rigid transformations.

I would say it makes many non-obvious facts into obvious ones. It’s just a really useful way to analyze problems, and as such it’s applicable in basically every area of math, as far as I’m aware.

 

[robphy]

Many problems in math are solved by trying to reformulate the problem
to be similar to a problem you have solved before.
Sometimes this reformulation is gotten by transforming the original problem.

Maybe you learned to solve a problem with a 30-60-90 triangle in some configuration.
If presented with a different 30-60-90 triangle with the same leg-lengths but oriented differently,
one could use transformations to re-orient your axes and get it into the form you first learned.
But what are those transformations? Hence, one studies those so-called rigid transformations.

They may seem obvious... but one aspect of mathematics is proving properties,
not just assuming that they are true.

(In special relativity, one uses a different measure of lengths and angles than in Euclidean geometry.
So, rigid transformations there are different in detail... but have some similarities.
Intuition is not as reliable ... hence we need to prove things before we can use them with confidence.)

More generally, how does one know when two objects are practically the same type of object?