need help

[candid bluff]

1. Ask your question and show the work you've done so far. If you've posted a screenshot of a question, specify which part you need help with.
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   +close
   

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[mint lion]

Hmm

Try to draw a horizontal line

Tangent to the grey circle

But at its top

Then the top triangle is just a scaled down version of the outer one

Which also tells you that the 2nd largest circle is a scaled down version of the grey one

By the same ratio

[forest crystal]

Do you know how such a circle can be constructed? That could help you determine the dimensions.

[supple owl]

I’m sorry but I don’t get it :/

[supple owl]

Yes

[mint lion]

Here is a simple cut

So do you agree that the top triangle is a scaled down version of the outer one?

[forest crystal]

How is the center of the circle determined, along what line?

Can you make triangles out of the lines that determine the midpoint of the inscribed circle that might be used to calculate the side-lengths of the triangle?

[supple owl]

Yes

[supple owl]

No

[supple owl]

But idk by which factor

[forest crystal]

That's why I asked my question, that's going to help you in determining that factor!

[supple owl]

How?

[forest crystal]

Can you tell me how you make an inscribed circle?

[supple owl]

An inscribed circle is a circle that’s trapped inside of a triangle

You need the radius

To draw it?

[forest crystal]

Yes

[supple owl]

Which is 1cm in this case

[forest crystal]

Ok but before that, how do you determine the centre of the circle?

Hint: bisector

[supple owl]

Ahhhh

Well there’s a formula to find out the radius of inscribed circles

Let me google it real quick

[forest crystal]

You don't necessarily need that, you can do simple trigonometry to determine the side-length of the triangle with the hint I gave

I will try to draw it out in a bit, I don't have paper atm

[supple owl]

It’s this

[supple owl]

The perpendicular bisectors all meet at the centre of the circle

I’ll draw the diagram sec

[forest crystal]

The triangles they draw in here can be used to calculate it in a much simpler way

[supple owl]

Something like that

[forest crystal]

YES

What triangle do you know at least two things about?

Including one length

[supple owl]

Ah I only know the heights

Of these triangles

Them being equal to the radius

[forest crystal]

Are you sure that's all you know

[supple owl]

Of the circle

[forest crystal]

What kind of triangle is the big one?

[supple owl]

Equilateral

[forest crystal]

What does that mean?

That means that what is the same?

[supple owl]

Equal sides

[forest crystal]

And what else?

[supple owl]

Equal sides and equal heights

And angles too

??

[forest crystal]

Bingo

So if all angles are the same and the sum of the angles being 180

What is a single angle

[supple owl]

Wait let me label them

This is getting confusing

[forest crystal]

3 angle=180, so 1 angle=?

Maybe draw it a little bigger

[supple owl]

60 degree

For a 30 60 90 triangle

[forest crystal]

Well done

So that means that the whole triangle has sides with length?

[supple owl]

1, 2 and sqrt3

Since x=1 in our case

Messy version

Cleaner version

[mint lion]

If you know that the upper triangle is a scaled down version of the larger one, then you know that the second smallest circle is scaled down with the same ratio

[supple owl]

How do I calculate the ratio you’re speaking of?

[mint lion]

using this figure

you know the squared height of the large triangle is (2root(3))² - root(3)² = 12 - 3 = 9 (Pythagoras)

so the height is 3

[forest crystal]

You're almost there

[mint lion]

but the height of the smaller triangle is 3 - the diameter of the circle

= 1

so the ratio is 1/3

the second smallest circle has a radius of 1 * ratio = 1/3

so what happens when you sum all their areas

[supple owl]

Yeah

[mint lion]

$\sum_{i=1}^{\infty} \pi (1 \times r^{i})^2$ where $r$ denotes the ratio

[supple owl]

How did you call ate the height of the smaller triangle ?

Calculate

[mint lion]

the height of the large one minus the diameter of the circle

you can see it on the cut i made

[blazing swallow]

Rion

[mint lion]

blue minus purple

[supple owl]

I get it

I get the ratio

[mint lion]

So yeah now you could try computing a geometric series maybe

[supple owl]

Between the two triangles

Alright

Thanks

🌻

[mint lion]

note that I only computed the area of all the smaller circles at the top

so you multiply that by 3 for all the other ones

and add the area of the bigger one

[supple owl]

Alrighty

[forest crystal]

They're first supposed to do the circumferences, but that's not that different tbh

[mint lion]

Oh yeah my bad

But yeah the idea works too