Ah, here it is 🙂 The welcome center i had no idea existed

Ok so hi i'm an energized panda

Uh I like Chopin. And number theory.

Cant wait for module 6.

@The-Blade-Dancer 🎉 🎉 Very nice work!! 🎉 🎉 Thanks for doing so much for the forum!! 🙂 🎉 🎉

@RZ923 Well I can play chess but I live in china so maybe a live one

Wow Thanks!

@victorioussheep Glad that you understood everything 😄

visiting old posts...

@fabulousgrizzly np! 🙂

@victorioussheep that would be like imo problem

find and prove the formula for the area of this figure

insert weird star here

@amusingminnow Good question! It's highly non-obvious, but the rules of operations say that when there are stacked powers with no parentheses, we work from the top down, so for example, 2477c580-a159-4f31-9eae-6ee1887a5d7a-image.png
We can continue this pattern to evaluate the entire expression.

(Sorry for the bad formatting lol)

@audrey Thank you so much, Audrey!

@victorioussheep Oh okay, sounds good! 😄

@professionalbronco No worries about the bump! Super glad to see you learning so much, and even coming back to correct an old post 🙂

You're absolutely right, these sums don't converge, so you can't manipulate them as if they were actual equations. In fact, \(1+2+3+4+...=-\frac{1}{12}\) is just really misleading in general. While \(-\frac{1}{12}\) is related to it, it's not really "equal" to it in the normal sense.

Great job once again!

Thanks @RZ923

ok ok...

lol recently there was a lot of people online, so maybe the system just couldn't take so much

@TSS-Graviser Yes, the isosceles triangle does look very different, but the neat thing about the underlying math here is that it doesn't matter what the triangles look like! This proof of the Exterior Angle Theorem just uses the fact that any triangle's angles add up to \(180^{\circ}.\) Whether it is acute, obtuse, right, isosceles, equilateral... every triangle's angles add up to \(180^{\circ}.\)

The angles that make up a line add up to \(180^{\circ}\) as well, and this is the reason behind the Exterior Angle Theorem. The sum of the interior angles of the triangle equals the sum of the angles on the line drawn by extending one of the triangle's sides. 🙂

@TSS-Graviser There isn't much of a big "reason", it's simply how we define the "measure" of an "arc".

If you think about it, arcs aren't really "angles", right? But, we find that it is very useful to just "say" that an arc has an "angle measure". But why, and what would it even represent?

An arc is just part of a circle. So, we can think of the "angle measure" of an arc as how much the arc goes around the circle. For example, the full circle is 360 degrees - if you stood on a playground and drew a full circle around you with a stick, you'd spin a full "360 degrees". If you only went a quarter of the way around, you only draw "90 degrees" of an arc.

As you can imagine, how much you go "around" is exactly the central angle you make!

72888ed9-7bcc-40ef-aef1-531f11fd12b0-image.png

So why is this useful? Off the top of my head, a good reason is that using "arcs" instead of "central angles" can be a good shortcut, and can be easier to think about. For example, inscribed angles along the same arc have the same measure!

db49cd76-307c-4cdd-b311-750e921f0e38-image.png

And we can say that the measure of each of those inscribed angles is half the measure of the intercepted arc, and we don't have to draw in the central angle.

@imadandelion no joke i jumped a little when i heard the music

@creativegoat Hi there! 🙂 If you've come directly from the course lessons, you don't see the "New Topic" button because you are already inside a post thread. In order to come out of your current thread, you can click on one of the parent categories listed on the top, which shows your location in the forum category structure.

For example, the "Home" on the top of this page will take you to the set of main categories:

forum-how-to-make-new-topic3.png

 
If you get out of your current thread and into a parent category (and any parent category will do), then you will see the "New Topic" button either at the bottom of the list of main categories, or at the top preceding the list of topics in a given category.

forum-how-to-make-new-topic.png

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I hope this helps, and, if you're new to the forum, then a very big welcome to you! I hope you find this a useful and fun place! 🎉 🎉

@amiabletiger Hello, welcome to the forum!