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

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• cbarker1
In summary: One way is to consider symmetries. For example, a right triangle is symmetric about the vertical and horizontal axes.In summary, cbarker1 is relearning high school geometry through Khan Academy and find it helpful to work with transformations, dilations, and symmetries. He believes this is because they make many non-obvious facts into obvious ones and is applicable in basically every area of math.
cbarker1
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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

cbarker1 said:
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.

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...

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• High School Geometry _ Khan Academy.pdf
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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.

cbarker1
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?

cbarker1

1. What are rigid transformations in geometry?

Rigid transformations are transformations that preserve the size and shape of a figure in geometry. This includes translations, rotations, and reflections.

2. How do you perform a translation in geometry?

To perform a translation, you need to move the figure a certain distance in a specific direction. This can be done by sliding the figure along a straight line without changing its orientation or size.

3. What is a rotation in geometry?

A rotation is a transformation in which a figure is turned around a fixed point called the center of rotation. This causes the figure to be in a different position but maintains its size and shape.

4. How do you identify congruent figures?

Two figures are congruent if they have the same size and shape. This means that they can be transformed into each other through a series of rigid transformations.

5. What is the difference between a reflection and a rotation?

A reflection is a transformation that flips a figure over a line, while a rotation involves turning a figure around a point. Both preserve the size and shape of the figure, but the direction of movement is different.

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