geometry-and-trigonometry

[fading crater]

I understand that 1/2 is because of the halved base, which is the opposite side.

[zealous pike]

Yes, anything halved is multiplied by 1/2. If the base was 2 before and was halved, we have 2(1/2) = 1

Let's say that this is an equilateral triangle that was halved (don't mind the ugly drawing)

[fading crater]

Okay, i understand that.

[zealous pike]

We define the opposite side of an angle as the side to where it is "looking at". Let's say an angle of 60°, its opposite side is sqrt(3). The opposite side of 90° is the hypotenuse itself, 2.

Now. When we have Sin(x) where x is any angle, we say that sin(x) is the ratio of the opposite side of the angle x by the hypotenuse. Do you follow?

[fading crater]

Yes.

[zealous pike]

Can you tell what is the value for sin 60°?

[topaz turtle]

uhh i don’t know the correct channel for this but can someone help me with my ixl

🥹🫩

[fading crater]

Pi/3

Wait.

[zealous pike]

Pi/3 is the same as 60°, but it isn't the value for sin 60°

[fading crater]

Hold up.

Sqrt(3)/2?

[zealous pike]

That's right

You got this

Now you know how the value for sin(30°) comes to be, right?

[fading crater]

Yeah, i think so.

1/2?

[zealous pike]

That's right

Summarizing.

A bit unrelated first, just for a clear of mind. We define a function as f(x) where x is the input for that function. Therefore, if we have f(x) = x and choose 5 for input value ak f(5), we have that f(5) = 5

Sine, cosine, cotangent, etc, falls into the same category: they are all functions (just aren't noted as f(x)). We can say that if we have sin(30°), sin(60°)… sin(x), , we are inputting values so that it will calculate the ratio of the opposite side of those angles by the hypotenuse of a certain triangle.

Most of the time, we don't know which triangles we are dealing with when we input let's say sin(2°)

But very good people have already computed some of those values for us today, which we recall by the trigonometric table

, we see that the sin(45°) = sqrt(2)/2. We can say that there is a triangle whose opposite side of 45° is sqrt(2) and the hypotenuse of that triangle is 2

Are you following?

[fading crater]

Yes.

[zealous pike]

An important thing to note is that we not always write the angles between the parentheses. It is common to let them without it

Sin(30°) = sin 30° = sin π/6

Coming back to your question. You got to the point where

Sin(3x) = 1/2

And you need to solve for x. Let's not do this for now and just call 3x by z. We have that 3x = z

This gives us

Sin(z) = 1/2

Ops

Can you tell what angle in radians or degrees, z needs to be so that sin(z) = 1/2?

You can forget that z = 3x here. Just think of z as any angle

[fading crater]

Maybe sin 30°?

So pi/6?

[zealous pike]

That's right

You have that z = pi/6 = 30°

We said that z = 3x. Therefore, we have z = 30° = pi/6 = 3x.

There is a real number x that if it is multiplied by 3, it becomes pi/6.

in mathematical words, it is the same as saying:

3x = pi/6

If we divide both sides by 3, to what result we arrive?

And what happens if we substitute that result into x and multiply it by 3?

[fading crater]

x = pi/18?

[zealous pike]

That's right

Now

Multiply it by 3

What do we have?

[fading crater]

Both sides?

[zealous pike]

Just pi/18

[fading crater]

Pi/6

[zealous pike]

Do you understand now how sin(3x) = 1/2?

[zealous pike]

You were right before 😭

It is by 3 not 1/2

[fading crater]

Sorry, registered it just now.

[zealous pike]

No problem

Do you understand now how sin(3x) = 1/2?

[fading crater]

It's because of sin 30°?

[zealous pike]

Yes

3x becomes 30° somehow

We discovered that how

We discovered what value is x, so we have… sin(3x) = 1/2

What value do we substitute for x?

[fading crater]

Pi/18?

[zealous pike]

No

[zealous pike]

Yes

That's right

Pi/18!

You have come to the conclusion that sin(3x) = sin(3[π/18]) = sin(π/6) = 1/2

You may be confused in some bits for now, but it may become rather clear as you think about all this explanation.

As an exercise to understand better, try:

Cos(6x) = sqrt(2)/2

Cos(πx) = sqrt(3)/2

Solve for x

[zealous pike]

Remember of these values

[fading crater]

1. x=pi/4
2. x=pi/6

[zealous pike]

Not quite.

You know the angles that give sqrt(2)/2 and sqrt(3)/2 for cosine

[fading crater]

1. 3*pi/2
2. 2*pi/6
   ?

[zealous pike]

Not yet but you are getting the heat

[fading crater]

For the first one.

Am i supposed to divide both sides by 6?

[zealous pike]

That's right

[fading crater]

So, sqrt(2)/12?

[zealous pike]

No

You want to divide both sides by 6 in another equation, one of which you just sent here

[fading crater]

I don't know.

[zealous pike]

You know

Think a little

[zealous pike]

Why do we want to find x? What's the purpose in that here?

What is the value for x called if we input it into cosine?

The answer for the last question is a word that starts with "a" and ends with "e"

[fading crater]

Angle?

So is x = pi/24?

[zealous pike]

YOU GOT THAT BRO

[fading crater]

Is that a yes?

[zealous pike]

Yes

[fading crater]

Ok, so the other one is pi/6*pi?

[zealous pike]

Yes

[fading crater]

Ok, got it.

[zealous pike]

[fading crater]

[zealous pike]

It took longer than needed but that's just how I like to explain stuff

[arctic hatch]

May I know will my teacher made the exam so hard? Anyone know?

[fading crater]

It's due to having to type out messages.

[zealous pike]

Indeed

[fading crater]

[zealous pike]

Type out, wait for a reply, wait for a reply of the reply, wait for a reply of the reply of the reply…

[fading crater]

Yeah...

Sorry for that.

[zealous pike]

Nooo

No be sorry

Be happy actually

I'd be pissed off only if I was unable to explain this

It's my job as someone who wants to share the understanding of what I understand, to understand the understanding of my listeners and work around that

[shrewd valley]

Can somebody please explain to me basic triangles

SAT on Saturday and I have no idea how to find lengths of sides or angles

Please just explain the gist of it. Math is not my strong suit whatsoever

[paper spear]

for modelling sinusoidal functions why do u divide 2π for the period

if it's getting bigger dont u multiply it

or atleast put it the other way

maybe idkk

[fading crater]

Thank you regardless!

[pliant bane]

I need help, i don't understand how to solve this
define line shadow of 2x-b=3y which is reflected on the line y=-5 with y=k
(x',y')=(x,2k-y) formula

[empty yew]

SAT is a good exam compared to JEE(sedly)

[spiral plover]

Wdym?

U don't need to know how to do that for the SAT

Just special right triangles should mostly do the work

[obsidian harness]

send the image of your problem

[obsidian harness]

you're forgetting that the SAT exam has been digital for a while now

before you could use your graphing display calculator

[spiral plover]

Yeah i know

😭

[obsidian harness]

yes then right triangles with any angles and sides can come up

[obsidian harness]

the period of sin(x) and cos(x) is 2π
now, you should think of sin(2x) and cos(2x) like the functions are playing on 2.0x speed

in other words, sin(2x) is a compression, 2 > 1
sin(x/2) is a horizontal stretch, 1/2 < 1

[spiral plover]

No

That's not something they test

[obsidian harness]

source?

https://www.reddit.com/r/Sat/comments/vjkgks/khan_academy_right_triangle_trigonometry_problem/

there's something as devilish as this for example

[spiral plover]

That's khan

Not an official sat question

[obsidian harness]

https://www.reddit.com/r/Sat/comments/hz7q03/right_triangle_trig_on_sat/

[spiral plover]

Nothing here contradicts what i said though

(3,4,5) and (5,12,13) is just pythag things

[obsidian harness]

okay you should have clarified

[spiral plover]

I've not seen a single question that directly tests you on something like that

[obsidian harness]

cause 'special right triangles' means only the 30-60-90 and the 45-45-90 angle ones

[obsidian harness]

doesn't mean it couldn't come up

I know that SAT doesn't have a predefined syllabus which complicates things

[spiral plover]

I assumed that the "focus" was more on the angles

[spiral plover]

No they reuse questions

[obsidian harness]

they could also make new ones

[spiral plover]

By that logic, a comp math question could also potentially come up because it's theoretically well within their syllabus

[spiral plover]

They're all reiterations of the same concept they're trying to reinforce

I've yet to see a novel question

[obsidian harness]

no, because they already test right-angled trig with respect to

1. special right triangles
2. integer side lengths right triangles

making the angle or a side 'not-neat' and expecting you to use Desmos to calculate the answer isn't novel

will these common styles be more likely to come up? absolutely

[spiral plover]

You're operating from a theoretical pov

😭

I would focus more on all the official questions

one has access to

A certain style of question is unlikely to show up

If it does then it's experimental

Look at the march sat

or the may one

actually look at the entire last year for reference

[obsidian harness]

where?

[spiral plover]

well there are many sources

telegram

bluebook+

I've seen it all

[obsidian harness]

fair okay

okay I see I was wrong

[shy grotto]

I doubt you're going to find one within the past few SATs to be honest

But iirc, I took november 2024 (quite old) and there isn't much "bad" triangles

[obsidian harness]

but yeah it does make more sense to test with a variable instead of random numbers

[obsidian harness]

which is what I was getting at earlier

[topaz turtle]

could i get some homework help?

[grave pond]

If you actually ask a question, there's a chance someond will answer it.

[topaz turtle]

Oh sorry

this is what i need help on

i’m not sure if this is the right channel for this though.

[buoyant island]

hey, is there someone who can teach me geometry, i am joining my highschool math team next year as a freshman and the paper they gave me has geometry on is. can someone help me learn geometry over the summer

[worn scroll]

https://tenor.com/view/unit-circle-butterfly-math-mathematics-can't-stop-thinking-gif-3156853134918236393

[obsidian harness]

if you haven't stared at the unit circle long enough to make you hallucinate, you're not a real mathematician

[edgy spoke]

I don't know for sure if I can help, but you can dm me if you have a question or a problem, I'll try to help if I can.

[empty yew]

Help here so that we can see that solution too

lol

[edgy spoke]

Sure, I will if I can and have time

[edgy spoke]

I thought they wanted a tutor or something

[empty yew]

But that's against server policy lol

[edgy spoke]

Oh, i must've missed it. I'm sorry, I was in a hurry

[grave pond]

It's not against server policy to help people in DMs. It is mildly discouraged to use the server to arrange paid tutoring, in DMs or elsewhere. (And strongly forbidden to advertise paid tutoring services out of nowhere).

[edgy spoke]

Thanks for clarification

[severe grove]

how many times do you think about the unit circle daily?

[grave pond]

Weird question. Who keeps count?

[trail tendon]

the more I think to count the more I think about it which makes the data corrupt 😔

[visual flume]

help pls

!status

[lime crown]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[visual flume]

1

[obsidian harness]

use barycentric coordinates

also this belongs in #competition-math

use this, and then the centroid must lie on this line

[hoary totem]

u think theres a way to solve this without coordinate geometry?

[obsidian harness]

barycentric coordinates aren't coord geo

garlic said to make the line an axis and use the fact that centroid is the average

idk what they mean

[hoary totem]

oh
wait that makes this question super easy

||make the line through G the x axis and G the origin||
||A is (-a, x)||
||B is (-b, -2)||
||C is (c, -5)||

||then because the centroid is the average of A, B, and C and is at the origin||
||(-a - b + c)/3 = 0 and (x - 2 - 5)/3 = 0||
||from the second equation we have x = 7||

it is coordinate geometry

[obsidian harness]

I mean that gives you the answer but doesn't show why it works regardless of the triangle or the line

but you figured it out yeah, I didn't expect that

[hoary totem]

how is it regardless of the triangle or the line

[obsidian harness]

that's my point exactly

[hoary totem]

if the line or the triangle changes the coordinates of A B C change

thus the length x also change

[obsidian harness]

yeah but you're stating the value of x doesn't change

[hoary totem]

it does what

[obsidian harness]

the question implies that x doesn't change

[hoary totem]

it doesnt at all

it gives the lengths 2 and 5 which lock the triangle and line to this specific configuration

a different line through G obviously gives different cooridnates for A B and C

[obsidian harness]

okay now I agree with you

my point was the coordinates change but the value of x remains the same

[obsidian harness]

but then you didn't show this fact

you just tested with one possible set of coordinates

[hoary totem]

my point was that x is the coordinate

[obsidian harness]

oh wait let me read

okay I misread what you wrote

[hoary totem]

x is the y coordinate of A under those specific axes

if the line changes the axes change

so the value of x changes

[obsidian harness]

ahhh okay so you did cover every possible setup up to rotation and translation so yeah

[hoary totem]

as seen here

[obsidian harness]

sorry

[hoary totem]

no need to be sorry lol

[obsidian harness]

if I read I wouldn't have needlessly accused you

I just skimmed it

[obsidian harness]

damn that's a clever problem

[hoary totem]

idk why the equation of the line was needed

[obsidian harness]

yeah you could just do "mean vertical displacement from line = 0, so x - 2 - 5 = 0" directly

[hoary totem]

true i was like what the hell but then as soon as u mentioned "the centroid is the average" i was like

o m g

[obsidian harness]

introducing coordinates confused me and you

anyways we got it!

[hoary totem]

yar

[obsidian harness]

thank garlic

I didn't actually say that

if I understood I would have figured it all out and told you earlier

[hoary totem]

-# whos garlic

[obsidian harness]

garlicbredfries

a fellow green helpful

[hoary totem]

i see!

[left mauve]

How did you make your text like that

[hoary totem]

-# text

-# text

[left mauve]

Ooh

Nice

[empty yew]

-# Hi shiru and executor

[regal kraken]

I belive this belongs here

tried to calculate k by converting cot in terms of tan

and tried to create a telescopic series

[empty yew]

Don't multipost clone!

[upper echo]

That's easy. First show that tan(pi/3+x)tan(pi/3-x)=tan(3x)/tan(x). Then it is just a telescoping product.

[regal kraken]

got it thanks cuh

[regal kraken]

ayy bro

you were the one who replied me in the help section right?

I am still stuck at that problem

[upper echo]

so what's the problem? cot=1/tan Bring it to the left side. You get products of tan(pi/3-x)tan(pi/3+x) which is tan(3x)/tan(x)

[regal kraken]

not this one

the other one you helped me with

[upper echo]

which one?

[livid quiver]

who can solve general formula for theta_? with given variables?

[crude quiver]

oh

[worthy eagle]

bro its not clear

[crude quiver]

yeah agree

[upper echo]

you can find theta_? by this formula, where all angles are in radians (d is just the ratio of radiuses, not that segment marked d in the figure). By the way, only 4 parameters are enough to find all others.

[exotic cloud]

theta _ out?

[wild nebula]

anyone kno whats gon be on the regents

[grave pond]

I'm not sure what "the regents" is, but that sounds a lot like you're asking for cheating help with something?

[left mauve]

Yeah prolly, regents is high school exams in new york

[serene hedge]

new york state high school exam

oh

he said that already

my fault

[trail tendon]

bro is visually deaf 👍

[serene hedge]

usually they just call that blind

[trail tendon]

thats what AVERAGE people call it

[swift gorge]

hi 😔 i'm super super super behind on school work and i need urgent help

i understand some things but lately i don't

[obsidian harness]

do you know what an angle bisector is?

[swift gorge]

ummm

is it the grey line?

sorry if i sound stupid im very bad at math it;s hard for me to pay attention to it

[obsidian harness]

yes

but what does that tell you about angles ACE and ECB, if CE is an angle bisector

[swift gorge]

theyre the same just flipped

so they have the same angles

[obsidian harness]

yep! they're equal angles

yeah so now comes the algebra part, well set up the equation first

[swift gorge]

is it 2x-15=x+5?

it is

it gave me 20 and that was the answer

umm hold on im gonna see how far i can get until i need help on stuff thank you south

ahh im not sure if this was the right channel either sorry 😭

[opaque radish]

Im having to relearn trig to take a math placement test and am a bit confused. I understand how to find the angle but im just not sure how they want me to estimate the number of revolutions? Should i just assume that in figure T2 Theta is 4/12?

[maiden brook]

This isn’t explained very well here

But no, it’s **-**1/3 * 360

[opaque radish]

isnt it based on the unit circle which is split up into 24ths?

[maiden brook]

huh?

[opaque radish]

nevermind i was thinking of something completely different

I thought the unit circle was something else

[visual flume]

help pls

[obsidian harness]

have you tried drawing it out?

also do you know what concave means? it's not just any pentagon where all 5 sides are equal

[empty yew]

,, \arccot(x)=\begin{cases}\arctan(\frac 1x) &x>0 \ \pi+\arctan(\frac 1x) &x<0\end{cases}

[somber coyote]

Monkey•D•Luffy

[empty yew]

What is it for arctan(x)?

[empty yew]

,, \arctan(x)=\begin{cases}\arccot(\frac 1x) &x>0 \ -\pi+\arccot(\frac 1x) &x<0\end{cases}

[somber coyote]

Monkey•D•Luffy

[empty yew]

Is this correct?

[left mauve]

Ye

[empty yew]

tq cutor.

.exe

[wispy void]

what would be the answer to 21 and22

i confused

[vague dome]

hi

Can anyone help for analytical geometry and trigonometry

[summer cradle]

since JL is a diameter which means it divides the circle into 2 equal halves

arc NJ = arc MJ

I think the question assumes that JL is perpendicular to MN

so JL bisects MN

then form triangle PKN

KN is radius

and so is KL

[obsidian harness]

or alternatively, angle MKN = 98
then angle LKN = angle MKN / 2 and then m(NJ) is easy

then as noted already, KL = KN (both radii) so you could:

1. split triangle LKN into two congruent right triangles, and then use right-angled trig
   don't forget to double the opposite side to the central angle to get LN
2. cosine rule also works too

[summer cradle]

98 is the arc length not the angle

[obsidian harness]

ah right, so angle MKN = 98

[sick zenith]

Omg is anybody online 😂 the geometry regeants is tmmr

[obsidian harness]

new yawker

[sick zenith]

How do y know- ou wait.. YEAHH

[obsidian harness]

only one state calls their exam the Regents

[sick zenith]

[sick zenith]

Eyyyy New York is just something

[left mauve]

There was a guy here just yesterday asking for what questions would be on the regents

[obsidian harness]

.......

[obsidian harness]

your bio screams chronically online.

[left mauve]

How so

[obsidian harness]

it just does

[left mauve]

Hm maybe because it has 3 different things on it idk lol

But yeah rn i am chronically online since I'm on holiday

[empty yew]

What does ask on server b4 DM means? lol

[left mauve]

Don't dm without asking on the server first? It's pretty clear

[empty yew]

before

i see

[south storm]

Hey anyone down later to help me to study for my final? Need to be locked in

[late fern]

hey guys idk which channel is appropriate for this but i'm trying to study for my A levels and struggle with understanding geometry, does anyone have any tips on how to get myself to understand the concepts of geometry? i'm solving questions regularly with help but can't do it on my own for some reason

[real willow]

A bit confused with this page in my math book

Help is appreciated

[maiden brook]

do u understand how to do any of them?

[obsidian harness]

they did q17 already btw

[obsidian harness]

for q18 you want to create another reflection (technically you need calculus to see why the shortest path is a reflection, but)

[maiden brook]

no u don’t, there’s a slick way to see it

[obsidian harness]

what's wrong with 'you want to create another reflection' ?

[maiden brook]

No I mean that u don’t need calculus to justify it’s the best

[obsidian harness]

and for q19, 20, recall that the shortest distance from a point to a line is the perpendicular distance

that should tell you what the special relationship between the two lines is

[obsidian harness]

oh really?

ah I know

yeah you just extend the line past y = 1

[obsidian harness]

that's how they figured out q17 btw, so props to them

[maiden brook]

I made a diagram for this before a while ago

https://cdn.discordapp.com/attachments/903463351789694986/1141065732684259368/image.png?ex=684a6a06&is=68491886&hm=825d54a1509585c427deae60fe09f5d33fbdb83d21b3c94d105af45edfa1233a&

[real willow]

I already got the rest of the questions btw :D
thanks for the help either way 🙏

[obsidian harness]

np!!

[edgy bone]

Hi. Can someone explain to me how cos(x) and sin(x) can be negative? arent they defined as follows:

cos: cos(alfa) = adj length / hyp length
sin: sin(alfa) = adj length / hyp length

Lengths cant be negative, so how can cos(alfa) < 0?

When I google I just see people mentioning the thing with the unit circle and how cos(alfa) goes between -1 and 1. but I still dont understand how it becomes negative and can only see it going between 0 and 1.

I feel like I'm missing something extremely basic here lol.

[exotic yarrow]

We're considering what I like to call "signed length"

,w graph x^2+y^2=1

[somber coyote]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by civil_service_pigeon

[exotic yarrow]

Ex. For this triangle, both legs "follow" the positive x and y axes, respectively, so we consider them to have positive length

but let's consider a triangle like this

[edgy bone]

Oh, is the definition taking direction into account and changing the sign?

[exotic yarrow]

again, the vertical leg "follows" the positive y axis, so we say that has positive length

but the horizontal leg "follows" the negative x axis, so we say that it has negative length

also note that the hypotenuse is always taken to have positive length

regardless of the quadrant

[edgy bone]

So the definitions i mentioned take the direction in the coord system into account for the opp and adj lengths?

[exotic yarrow]

sure

[edgy bone]

ah, then it makes sense. i just thought it was defined as a length, and a length can never be negative, right?

[exotic yarrow]

you can take lengths to be negative when convenient (where sign is based on direction)

Example - See solution 1 here: https://artofproblemsolving.com/wiki/index.php/2012_AIME_II_Problems/Problem_15?srsltid=AfmBOooMPUOj8W4Oycs7V_fe4EjjgMTU8J0xfVLgoH7u8GiOuPg6_D2_

[edgy bone]

aha okay. i didnt know that. thank you. now the definitions make sense

oh, signed distance exists. ive heard of signed distance fields, i shouldve figured

[fallen burrow]

Hi!

I'm looking for videos or resources to really comprehend trigonometry

[maiden brook]

On what level

[real willow]

does this look right 😓

[indigo nexus]

Which one ?

[glacial ember]

i think it does

but id prob change your reasoning for 21

[real willow]

the whole page in general

[fallen burrow]

I'm in high school, but I feel I'm just following orders. Why does knowing one side and one angle give all the other information about a triangle?

[dark sparrow]

one side and one angle?

are you sure about this?

it takes three elements to fully determine a triangle, not two.

[maiden brook]

And this isn’t so true even

[dark sparrow]

are you about to bring up the ambiguous case in SSA

or what

[maiden brook]

yeah

[fallen burrow]

it was an example

I think it is two angles (because the third is always 90 degrees) or one angle and one side

[spiral lodge]

Well, you're talking about right triangles then 😅

[signal vapor]

???

[zealous pike]

Aren't points 3 and 4 saying the same thing but in a different way? I can't note a difference in meaning

point 3 says that three points A, B and C not in the same straight line always completely determine a plane.

point 4 says that any three points A, B and C ofa plane, which do not lie in the same line, completely determine a plane.

Key differences: there is the word "any" in point 4. There is not the word "always" in point 4.

Here, does the word "any" denotes a set of three different points not in the same line or a set of points where one or more of the points A, B and C aren't in the same line, but two points can be in the same line? I came to the conclusion that his would be the difference, but I feel it is a wrong reasoning. Would someone give me a light?

[dark sparrow]

Aren't points 3 and 4 saying the same thing but in a different way?
nope

1.3 says that for any three non-collinear points A, B, C there exists a plane alpha that contains them all

1.4 says that this plane is unique i.e. there's one and only one

[zealous pike]

I see. Thank you

[mellow pollen]

i have a question from my a level exam today
for sin(nx)=k (0<|k|<1) (0<x<2pi) to have six solution, what is the value of n

n=3 right?

[spiral lodge]

Yep

[mellow pollen]

alr thanks

[obsidian harness]

can check using Desmos

[mellow pollen]

yeah i verified

thank u tho

[visual flume]

help pls

i got da and fb down but idk what to do from here

[indigo nexus]

Ed is the same

And the f is x

[obsidian harness]

call the intersection of DA and FB point P

then use the sine rule on triangle GPB

[obsidian harness]

what?

[indigo nexus]

Ed is the same as ha

[obsidian harness]

ok

[obsidian harness]

ah actually, might be better to use triangle GHP

[signal vapor]

This is one of those questions where coordinate geometry looks to be faster

[obsidian harness]

yeah that's the right triangle to use

well I mean all of them are a bit iffy

[restive sphinx]

Why equation of the line in normal form is given by x cos θ + y sin θ = p?

[echo beacon]

is it root3

[visual flume]

yeah

[echo beacon]

||pba is an isosceles triangle||

||an octagon is 135 degrees per angle||

after that its js spamming angles

no sine rule or anytying needed

[visual flume]

yeah

[visual flume]

i smh keep getting x = 120

but that doesnt look correct

wait it works when i plug it in with other shapes

bruh

idt its correct tho

[echo beacon]

what exactly are you trying to do

[blissful storm]

"A vertex of a triangle lies on the circle passing through the midpoints of the adjacent sides and the barry center. Compute the median drawn from this vertex if the opposite side has length a"
How do i approach solving this problem?

[echo beacon]

hint: ||start by proving that PBA is isosceles||
if u can do that i'm sure the problem's gonna be really easy for u

but if you're trying something urself and getting it wrong, i personally encourage u to keep going until u find out whatever it is that you're doing wrong because that's (IMO) much more important

[visual flume]

ohhhh wait i changed symbols wrong

[visual flume]

pba?

[echo beacon]

yes, where P is the intersection of DA and FB as south said. sorry for not clarifying

[visual flume]

oh wait sry for no context this is another problem

mb

[onyx marlin]

do you learn about topology/manifolds in pre-uni geometry?

[hoary prism]

no usually it is about like euclidean plane geometry

[lofty spear]

[upper karma]

this

[sage radish]

you can use circles

[upper karma]

?

[white cloud]

Is it not just angle chasing

Let measure of angle A be x

AED is then x

[rancid dragon]

Hello Humans, I want to deepen my understandings of trigonometry while making sure that I am covering enoug htopics from the standard 11th 12th Trigonometry syllabus (well, you can compare it with the Indian syllabus for intuition).
I am currently thinking of beginning "Plane Trigonometry" by S.L. Loney. I am kind of confused, because the book is from 1893. Should I continue with the book? I know Trigonometry doesn't get yearly updates like iOS. But still, is it compatible with modern trigonometry practices and similar?

[grave pond]

Old mathematics does not suddenly become false. The largest risk with an old book is that it will have a focus that's not all that relevant anymore. In particular, a book from that timeframe may place a lot of importance on clever tricks that would have been practically necessary to compute numerical results for problems where today we would just chuck everything into a coordinate system and throw a computer at it.

Of course, clever tricks will never entirely go out of fashion -- it's more a matter of weight and focus than a matter of "this is now completely obsolete".

[ember scroll]

I have a question, it is possible to know if two squares of different sizes, ranging from point A to point B at a certain constant speed, collide at some point, and know the position where the squares would be when colliding?

[unkempt topaz]

do the lines represent trails, or where the boxes are about to go?

[ember scroll]

Lines represent the Start and the End of the Box movement.

[ember scroll]

☝️

[unkempt topaz]

so are the boxes at the start right now?

[ember scroll]

yes sir

[unkempt topaz]

so the first thing im thinking of is, pick one box and view its perspective

so like, instead of the green box going left by 50, you keep the green box still and make the blue box also go right by 50 as well as its current motion

but then im afraid its probably not going to be a fun solution, just a lot of line formulas

and seeing the trajectory of the lines

[ember scroll]

I forgot to mention something, Speed ​​means how many squares each frame moves, Speed ​​= 20 means that you move 20 squares in 1fps, so 1fps would be like 1 second.

[ember scroll]

Sorry, I didn't get the idea.

[unkempt topaz]

if youre in a car going 20mph, and theres a car going the other way at 20mph, then from your perspective the car is passing you at 40mph

does that make sense

[ember scroll]

Hmm, but that's only true if, for example, I'm going from bottom left to top right. And the other car is going from top right to bottom left.
In this case, one car is going from bottom left to top right, and another is going from top right to left. So the speed wouldn't be the sum of both. In any case, I'm not sure how that would help determine if when both squares moving at different speeds they collide in some point.

[unkempt topaz]

correct, i deliberately used a simplified example

i am assuming, you know how fast the box is moving upwards, and also how fast its moving horizontally?

so then you can just modify the horizontal speed

[ember scroll]

The blue is 20 upwards and 20 horizontally. Although I still don't know how to tell if the two collide. 😅

[unkempt topaz]

so we can make that 20 upwards and 70 horizontally, and keep the blue one still

but then its going to be a lot of checking points, I'd recommend finding formulas of where the corners are going to travel first

but I can't imagine a neat formula

(becuase it would also depend on the size of the squares)

[maiden brook]

for geometry in particular, older books might be better cause geometry was more heavily emphasized in the past?

[grave pond]

That's a possible consideration, if you're reading out of interest rather than being strictly focused on useful knowledge. :-)

[upper karma]

i need help

[tardy vector]

you know that abc and cde are equilateral triangles

so from there you can work out ||angle ace and angle acd||

and using that info you can think about ||similar triangles||

[upper karma]

hmm i tried that but still did not get it

[dense karma]

Equilateral triangles are also equiangular, and all triangles must add up to 180 degrees, if that helps

[upper karma]

how>

[dense karma]

How to which part?

All similar triangles have congruent angles.

[grave pond]

They're trying to tell you that, for example ∠ECD = 60°.
Is that what you're having trouble with, or is it later in the instructions?

[dense karma]

Because
A.) all equilateral triangles are equiangular and
B.) all equilateral triangles are similar, and
C.) all triangles add up to 180 degrees, and
D.) all similar triangles have congruent corresponding angles

Then each angle from the triangle must be sixty degrees, because 180/3 is 60.

From there, you can work out that angle ACB is 60, like all the angles in triangle ABC.

60-41=19, so angle ACE is 19.

From there, as @tardy vector said, you can use similar triangles to solve.

[jovial tangle]

I noticed the angles of bce are same as the angles acd and since d is equal to 60 degrees then x = 32 degrees and take a closer look of acd and bce triangles slide lengths.

[final frost]

I literally solved everything angle and I was looking for congruence or isosceles triangle but then I gave up and then I saw the answer response, I was literally so close. In geometry you always gotta pay attention to the lil details

Bc using SAS congruence you can prove those trinagles are congruent and since it overlaps with an equilateral trinagle you subtract 60 from 92 and you get 32

[obsidian harness]

<@&268886789983436800>

[zealous pike]

Why does it say "almost identical" if the proof is fairly identical?

If L1 and L2 are supplements of L3, then L1 + L3 = 180deg and L2 + L3 = 180°, therefore, L1 = L2

L1 + L3 = 180deg
L2 + L3 = 180°
L1 + L3 = L2 + L3
By subtraction,
L1 = L2

The only thing that changes is the size of the angle

[dark sparrow]

yeah that's why it says almost

there is only one difference

180 instead of 90

[zealous pike]

So mathematics is that strict with differences

[empty yew]

Mathematics is always strict! We are the ones who take it rigorous or not for our cases!

[maiden brook]

?

[empty yew]

most importantly You can't say mathematics isn't mathematizing lol

We are the ones who don't understand it clearly if it seems to be wrong!

[zealous pike]

that's an interesting philosophy

So, if the steps for proving two theorems are the same but the value of something in one theorem is different from the other, we say that the proof is almost identical?

[empty yew]

The book that mentions conveying that

[zealous pike]

Can we say that the proofs are similar?

[empty yew]

Why are you this much obssessed with "almost" intsead of the theorem? lol

[empty yew]

sure!

[zealous pike]

I just have nothing to do

Im bored so Im brainrotting

[empty yew]

This should be used by me to you lmao

[empty yew]

Also proofs are important in higher level of math

Good Insight to look at the proofs detail lmao

[zealous pike]

I like trying to prove or demonstrate something

[empty yew]

https://en.wikipedia.org/wiki/Mathematical_proof

[empty yew]

Prove something useful! Good Luck!

[zealous pike]

Although I have almost no basis in the methods of proving. So far, I've only skimmed through direct proof, contradiction/absurd and that one for natural numbers

[empty yew]

I am eager to look at your proof If I can understand lol

[empty yew]

Me too

[zealous pike]

Induction

[zealous pike]

Direct proof is basically reaching a conclusion through already accepted ideas. Can we say that mathematical demonstrations (like reaching the quadratic formula) are a form of direct proof?

[empty yew]

I think so

Also I admit that I am no more than high schooler

[wet pivot]

Any good book of geometry

I m a newbie

[zealous pike]

Me too. Despite the fsct that I'm 22yo already

[zealous pike]

This one I'm reading is good. "Geometry: seeing, doing and understanding"

[signal vapor]

you can try Euclid's elements

||ignore the statement above any new geometry book works||

[zealous pike]

Euclid's elements works too

But

You'll want to vry

Cry

Maybe

[signal vapor]

it isnt recommend

i learnt it the hard way

you can try egmo

[final frost]

What does that mean angle one plus angle 2 =90 and angle 3 is 90

[final frost]

O nvm I get it

[zealous pike]

Not necessarily. Let's say that we have an angle of L1 = 30°. There is only one value that is complement of L1, namely 60°

If we have the information that L1 = 30° and L2 and L3 are complementary angles of L1, then L2 = L3 = 60°, because no two different values sum up with 30° to become 90°

Same goes for supplementary angles

[final frost]

Ok

O it was worded differently

My bad

[maiden brook]

bro this isn’t that deep

[elfin dove]

help

Help

I’m new to geometry

I want to know what the sign means

[cunning lion]

$\implies$ means ``implies''

[somber coyote]

cloud

[elfin dove]

can you show me a example of it

I just haven’t learn that yet

[cunning lion]

a shape being a square implies it is a rectangle

[final frost]

It means like if then

If p then q

I think

[elfin dove]

false?

I’m not sure

[final frost]

Like if both p and q are false then the thing is true but other than that mostly the “then” part is the answer

[cunning lion]

for "P implies Q" to be true, we have either:

• P is true and Q is true, or
• P is false (it doesn't matter what Q is then, because the statement doesn't say anything about Q then)
  if P is true and Q is false, that's the only situation where "P implies Q" is false

[elfin dove]

ooh

[final frost]

Wait what

[elfin dove]

What happens when it has the not operation sign

the ~

wait it like the same thing

[cunning lion]

well you apply the same logic just bearing in mind that not p and not q have the opposite truth values than p and q

[elfin dove]

can someone double check?

I’m not too sure

[cunning lion]

seems fine

[elfin dove]

yes thank youuuu

[empty yew]

Is logic a part of geometry?

[zealous pike]

ig it is part of everything

some geometry books do starts with logic though

[empty yew]

[cunning lion]

not really, but geometry classes often double as introductory proof-writing classes

[empty yew]

I see

How the derivative of volume of sphere equal to surface area of sphere?

Is this even make sense?

Is this a coincidence or has some intuitive meaning?

[cunning lion]

you can imagine. making a riemann sum of thin spherical shells of radius r and thickness Δr. then the volume of each thin shell is approximately surface area * thickness (this becomes precise in the limit)

so
[ V = \lim_{n \to \infty} \sum_{i = 1}^n S(r_i) \adif r_i = \int_0^R S(r) \odif r ]
where $S(r)$ gives the surface area of a sphere of radius $r$

[somber coyote]

cloud

[empty yew]

Whoaaaaaa

I see

That's why the derivative gives us the infinitesimal change or something

Is this applicable to any geometrical or even non-geometrical shapes? @cunning lion

[dreamy ferry]

Y'all uhm how do you prove a line is a median? Theres a chance I've forgotten

[dark sparrow]

median like in a triangle?

it needs to pass through (or start at) one vertex, and pass through the midpoint of the side opposite said vertex.

or to say it more snarkily: to prove something is a median, you have to prove it satisfies the definition of a median

[dreamy ferry]

Yeah, thanks
but if the vertex is right angled, then its equal to half of the opposing side, right?

[dark sparrow]

if your triangle is right and you're drawing a median to the hypotenuse it'll be equal to half the hypotenuse yes

[zealous pike]

If we are given a right triangle and draw a line bisecting the 90° angle, does it always intersect the midpoint of the hypotenuse?

[zealous pike]

[zealous pike]

Does anyone have any tips on how to find which quadrant they are in? So far, I've been comparing them with pi/2 and 3pi/2, but it is too slow

[left mauve]

Well that is the only way lol, if you want it to be faster you can convert them to decimal and see where they lie wrt 0.5pi,pi,1.5pi,2pi

[main spindle]

How do I find the coordinates of the vertices in this 3-dimensional figure?

[zealous pike]

to be faaster then it is only through repetition and experience then

[zealous pike]

you can think about the length of the sides as the terminal points i,e the values of the coordinates

for example, A would be (0, 0, 0). B would be (0, 3, 0)

[main spindle]

Understood, thanks
To clarify, D would be (5, 0, 0)?

[zealous pike]

[zealous pike]

what's the purpose of this one?

[cunning lion]

you put the point on the circle like you would a number line

[zealous pike]

But

there is no real application for that right? Like, the point of y for x = 0,5 in the unit circle is sqrt(0,75) not 0,8?

[cunning lion]

what?

[zealous pike]

The answer for 61 is (0.5, 0.8)

[cunning lion]

,calc cos(1)

[somber coyote]

Result:

0.54030230586814

[cunning lion]

,calc sin(1)

[somber coyote]

Result:

0.8414709848079

[cunning lion]

those are the actual coordinates. so it's an approximation good to 1 decimal place

[zealous pike]

I also got a very good approximation here for y? 😭

[zealous pike]

(x, y) is on the unit circle iff x² + y² = 1.

If x = 0.5 then y = sqrt(1 - 0.5²),
y = sqrt(0.75)

Isn't this correct also?

[cunning lion]

no, because the x-coordinate isn't exactly 0.5, so you can't claim to have the exact answer for the y coordinate

[zealous pike]

But it didn't even showed about cos or sin thus far, so I'm considering the exact numbers 😭

[cunning lion]

yes sure

but the point of the exercise is just to get approximate numbers based on the graph, and both numbers you get are necessary approximate with that method

[zealous pike]

And the book gave as the answer (0.5, 0.8) which we know they aren't exact numbers thoughh?
Y would I be wrong 😭

[cunning lion]

the only reason to give y in the form sqrt(0.75) is if that's the "exact answer". if it's an approximation then it would be more appropriate just to give it as a decimal

[zealous pike]

Oke

But

Not considering that exercise. It is right to say that (0.5, sqrt(0.75)) is on the unit circle right?

[cunning lion]

that is correct

[golden sigil]

can one or two sided polygons exist in geometry?

[cunning lion]

not in euclidean (flat) geometry, but yes in other geometry systems, for example in spherical geometry

[obsidian harness]

for example, in spherical geometry, the definition of a "line" (shortest path between two points) is a segment of a great circle

if you take the full great circle that splits the sphere in half, that's a polygon with only one side

[golden sigil]

oh okay

[wet python]

hi

[zealous pike]

The question asks me to rearrange sint_1, sint_2 and sint_3 in increasing order. That would be sint_3, sin_t1 and sint_2?

The reasoning is that any negative number is always smaller than a positive number. Sin(t) is negative if it is "under" the x axis, therefore, sint_3 is the smallest value. Sint_1 is the value most close to the x axis than sint_2, therefore, sint_3 < sint_1 < sint_3

Is the reasoning right?

I wish my keyboard had subscript letters

[dark sparrow]

sin(t_3) < sin(t_1) < sin(t_2) is correct

but sint_1 looks ugly without brackets

[zealous pike]

I agree

[blissful mason]

just finished my OCR additional maths qualification and this was the final question. could someone smart attempt it and see what they get? 🫡

also the circle hits the x axis at 0,0 ignore my slightly off diagram

or should i post this in help forum

i didnt see that lol

[grave pond]

So the center lies on the y-axis, and also on the perpendicular bisector of the line from (0,0) to (8,16).

[blissful mason]

it hits the x axis but the centres of the circle and squares are different

[grave pond]

The center of the circle definitely doesn't lie on the x-axis.

[blissful mason]

no the centre is unknown

the circle hits the x axis

but the centre is on the y axis

[cunning lion]

the brute force way to do it is to set it up as a system of equations in two unknowns (center and radius) and two equations (two known points on the circle)

[blissful mason]

i managed to get an answer

but i did a ton of steps so idk if i did it the right way

it was only 4 marks lol

[grave pond]

Instead of my suggestion above with "perpendicular bisector" you can also just say that the center is at (0,r) and then set up the equation r² = 8² + (16-r)².
Multiply out, cancel the r² terms on both sides, and you get a linear equation that you can solve for r.

(which ought to eventually give ||r=10||).

[blissful mason]

ah yea thats a much easier way than i did 😭

i used Pythagoras to find the top left corner to the point where it hits the x axis and then tan to find the angle it creates. then i could find the internal angle and use the reverse cosine rule to find the radius

i ended up getting 10 but it took me like 20 mins lol

thanks anyway

[serene umbra]

what is an OCR?

[blissful mason]

It’s an exam board

[serene umbra]

thank you

[serene umbra]

which grade are this exercices for?

[blissful mason]

Free standing qualification

Basically half an A level

it’s taken the same time as GCSEs

[serene umbra]

thank you

[urban surge]

Can someone pleasee help me with this Geo problem:

[lime crown]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[urban surge]

1

[sacred haven]

bot ghosting u💀 🙏 🔥

[serene umbra]

you can use the theorem of thales

[urban surge]

Can you help me use it idk what it is

[sacred haven]

how yall got so smart tho fr

[serene umbra]

ok

do you want me to give you some hint

hints

[urban surge]

yes please

[serene umbra]

so

[urban surge]

I asked my sister the same problem, can you also tell me the solution you got so that I can compare the answers

[serene umbra]

EC/ FG = FG / AB

[serene umbra]

OK

I found 10

[serene umbra]

FG = 10

[urban surge]

the system is telling me that the answer is a fraction

[serene umbra]

thenk i probably made a mistake 😕

[urban surge]

Can you please try this one :

[serene umbra]

what was the answer : - (

[serene umbra]

ok

[urban surge]

it was 80/7

[urban surge]

Have you found the answer

[serene umbra]

not yet

how about you

[urban surge]

I found that the side of the equilateral triangles is 2/sqrt3 i think

[serene umbra]

same

i found ( 28-3sqrt(3) )/24

which is 0.95016

@urban surge what did you find

[fast bronze]

does anyone have any practice tests

for skipping geometr

im in fisd

[maiden brook]

honestly coord geo isn’t too bad

[urban surge]

can you solve it

[maiden brook]

yes

[cloud oak]

hai, may i ask for help with geometry and measurement?

[glad dagger]

Is it wise to take geometry over the summer? A lot of people tell me yes, and a lot of people tell me not to, so idk, I'm not sure

[obsidian harness]

You'll have taken algebra 1 right? Before summer

[obsidian harness]

!da2a

[lime crown]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[glad dagger]

yes, I will have

[neon raven]

That looks like Thales

[neon raven]

Yeah Thales is enough to find the side length ratios, we don't even need to find the side lengths of the triangle

[thick talon]

guys i need help please

idk if i can post it here but i would like to ask how do i solve SPL = lateral surface area and Sp = area of the base for a prism cuz idk how does it work

if someone have a time to explain

[last bane]

I love trig etc idk if im going to rember al the rules and if I will do good in the exam

[short lynx]

i honestly havent seen anything else in trig other than sin cos and tan, does it even exceed beyond that

[spiral lodge]

Sure

sec and cosec

[grave pond]

I think that depends a lot on country. America appears to have a tradtion of presenting also sec, cosec, cotan.

[spiral lodge]

(but not in all countries, such as Italy)

[spiral lodge]

Yeah you anticipated me 👍

[grave pond]

I'm not sure there's anywhere weirder combinations like versines or exsecants are routinely taught today.

[short lynx]

i got 100% on my quiz

pro

[short lynx]

bro the pre-test was so much more harder than the actual test

💀

[left mauve]

Lmfao that is the way it goes

Scare you shitless and give you an easy paper

[viscid nimbus]

wtf

[burnt grail]

Hi I am taking Geometry when school starts, is it hard?

[left mauve]

No

[upper karma]

Hlo guyz

I am new to this server

[left mauve]

Nice, welcome!

[upper karma]

Is anyone here in class 11th? With maths?

Or am I only the youngest one 😦

[left mauve]

Lots of young learners here

[dense karma]

In my opinion, it’s not too bad! It’s easier than algebra if you’re a visual learner

[jovial bloom]

I had a hard time in trig and I ended up dropping the class for summer 2. I really need help to pass precac

The identities and forming them give me problems. I want to educate myself and be prepared for this time

[burnt grail]

Yay

[burnt grail]

I just finish freshman year

[neon raven]

👁️

[ruby oasis]

Hints?

[civic rock]

draw out the radiuses of the blue circle and segments from the center to A,B,C and D

you should be able to find similar triangles that can help you solve the problem

[twin depot]

Incoming freshman doing algebra 2 rn

[upper karma]

Oh

[teal zephyr]

guys how would u refute the staircase paradox?

[mystic umbra]

you could make right angle triangles given that the sides are tangential

might not be that useful... on contemplation

[lunar hull]

We have two things going on here. We have the sequence of curves, each of which has an associated length, and we have the limiting curve which has its own length. The limit of the sequence of lengths is not the length of the limit of the curves. These two things do not need to be equal and are not in the pathological examples in the staircase paradox.

[neon raven]

Yep, connect some lines and there are similar triangles.

[summer cradle]

what are you asked to do?

[neon raven]

it's likely to find something related to the incircle, probably the radius

[hoary totem]

how would i remove the parameter from this

[mystic umbra]

explain please

or i might be too underleveled to understand, one or the other

[left mauve]

Gotta eliminate t

[neon raven]

maybe try to find sin(t) and cos(t) individually because it's a system of linear equations? tbh I'm not really sure if the other stuff affects it

[viscid nimbus]

find fb noting that it is congruent to ef (pythagorean theorem) and solve for cf using the pythagorean theorem (bc is given)

[neon raven]

That looks like it can be dealt with using coordinates. But otherwise drop an altitude from F to CD and it should be the same.

[neon raven]

Oh yeah this works too

[visual flume]

i gotchu

hold on a sec

[urban surge]

ok thank you

[visual flume]

i swear ive done it in my aops hw before lemme check

[urban surge]

yeah i got this from aops

[maiden brook]

u can find FB and BC and then use law of cosines

[visual flume]

idt ur supposed to use any trig for this

gah i cant find it

imma js do it again

fbc is a triangle. draw an altitude from b to fc, and do some angle chasing and see if you have any special triangles from fbc

also help pls

[civic rock]

PQRS is a rectangle?

[visual flume]

yeah

i wish it was a square

;-;

[neon raven]

I would love to abuse symmetry on this one
Edit: doesn't really work.

[neon raven]

Oh, there's some similar triangles

Try triangles ||APF and FSD||

[civic rock]

oh nice find

I should've played around more with the angles

that just solves the whole thing completely

[viscid nimbus]

it would be so awesome if you could say af and cd are parallel to each other

[viscid nimbus]

looks like a pythagorean theorem proof with extra steps

[upper echo]

yep, you get the answer from this in one line.
bc their scale factor is just tan(30)

so, if x is 1/6 of the hexagon then (25+x)/20=3. Thus, [ABCDEF]=6x=6*35=210

[exotic cloud]

160

nah my bad, it may be a little higher than 160

[exotic cloud]

270

[upper echo]

nope, that's not correct. FP≠PA, so, the first line is false.

[exotic cloud]

How do you know FP≠ PA it looks like they are the same

[neon raven]

why are they the same :)

[left mauve]

"They look equal" is hardly an argument.

[exotic cloud]

idk

[civic rock]

the probability of two random segments equal to each other is practically 0

you just can't assume that

[summer cradle]

FD = ½x•sqrt(3) * 2 = x•sqrt(3)

APF ~ FSD

Sides ratio = FD / x = x•sqrt(3) / x = sqrt(3)

Area ratio = (sqrt(3))² = 3

Area of FSD = 20 * 3 = 60

Area of FED = Area of FSD - 25

Area FED = 60 - 25 = 35

½ * x•sqrt(3) * ½x = 35

x² = 140 / sqrt(3)

Area of hexagon = 3x²sqrt(3)/2

Area of hexagon = 210

[civic rock]

you don't need area of hexagon formula for this

just note that the triangles created by the hexagon's center and its sides are equilateral

[summer cradle]

yes Ik

[civic rock]

anyways can anyone help me with this problem:
S.ABC is a tetraheron. D, E and F lies on the medians SM, SN and SP of triangle SBC, SAC, SAB so that SD/SM=1/2, SE/SN=2/3, SF/SP=3/5. The plane (DEF) cuts the tetrahedron into 2 parts. Find the ratio of the volume of the big part/small part.

[summer cradle]

oh you can just directly do 35 * 6 = 210

[summer cradle]

which 2 parts?

[civic rock]

DEF forms a plane

that cuts through the tetrahedron

you have to draw the intersection of the plane and the tetrahedron

[summer cradle]

like this?

[civic rock]

yep

[neon raven]

These kinds of problems scare me

[civic rock]

yet it appears on my TSA training

[neon raven]

:(

I barely know anything about 3d geometry 😭

[civic rock]

oh what grade are you in

[neon raven]

9 :<

[civic rock]

high school this year?

[neon raven]

ye

[civic rock]

accidentally saw you're vietnamese as well so

[neon raven]

ok

[exotic cloud]

OHHHHH yeah right right , nice

[exotic cloud]

How is even possible that BM = MC, and those others ,if , for example, SD/SM=1/2 and SF/SP= 3/5

i don't understand

[neon raven]

It's just placing a bunch of points, I don't see anything weird lol

[civic rock]

true...

[exotic cloud]

oh fk

yeah sorry

but it isn't a regular tetraheron

[visual flume]

tysm

[summer cradle]

np

[civic rock]

oh my god I solved it

that took me so long

I'm so tired

you have to use this fact

and then the rest is easy

[reef void]

what theorem is that?

[civic rock]

idk what it is called

[north lynx]

probably a consequence of the "middle theorem" (that's what it's called in French)

[civic rock]

no

[wispy whale]

i'd like to say its appolonius theorem but it looks way too different

its probably just some random proportionality theorem

[civic rock]

I think I've seen it only once when I was reading about Apollonius' circle problem

[wispy whale]

actually now that you mention it

it does look like apollonius theorem

you just simplified it

[civic rock]

I think this has like nearly 0 practicality in problems that require proofs

I might be dead wrong

[wispy whale]

lmao, the only reason i recognized it was cause i studied alot of theorems for olympiads

[civic rock]

how do you even analyze those ratios then

without intercept theorem

[wispy whale]

hmm

[civic rock]

you can probably go a bit furthur and apply Stewart's theorem but the proof would be inelegant

[wispy whale]

yea no it would be really taxing to prove

ive had my fair share of problems like these

and would not like to go back

[civic rock]

idk if this is considered a consequence of intercept theorem or not

[wispy whale]

that is thales 3 parallel line theorem

[civic rock]

yea

[wispy whale]

where are you getting these problems?

[civic rock]

you form a parallelogram with that

[wispy whale]

i see

[civic rock]

my TSA training

[wispy whale]

oh i see

you're becoming a pilot then?

[civic rock]

no

TSA is vietnamese version of SAT

[wispy whale]

oh

wait you're in uni?

[civic rock]

"Thinking Skills Acessment"

[civic rock]

about to

[wispy whale]

i see

goodluck man 🫡

[civic rock]

I'm having gaokao next week

lmao

[wispy whale]

dear

sounds like a rough week for you

[civic rock]

nah gaokao here is not as crazy as in china

[wispy whale]

😭

oh i see

i based it off the china gaokao

and based off the stories ive heard

[civic rock]

my TSA score is already high so gaokao score won't matter much

[wispy whale]

i wouldnt say its pleasant in the least

[wispy whale]

damn, nice

i dont even know what uni i should pick

i should decide soon

[empty yew]

How sending big?

[left mauve]

It's a sticker bro

[worthy eagle]

idk if I should continue this catacy

[left mauve]

You broke it already

[visual flume]

absoutley do not use herons formula btw

[left mauve]

Is this a challenge?

[visual flume]

yeah

[left mauve]

Challenges aren't allowed

#prealg-and-algebra message

[civic rock]

can I post my solution to that here

[lime crown]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[civic rock]

or at least some hint

[neon raven]

yeah I thought about this too, but if this is the only way then I'm so done 🙏

[civic rock]

ye it involves addition of fractions which is not that practical in asian geometry exercises

proofs cough cough

[neon raven]

...well ig it's not so bad when you can find the side length ratios easily lmao

[upper echo]

You can rotate the triangle DCB around D so that C->E. Then the whole area is just two equal triangles with area 5*5/2 each.

[obsidian harness]

You sure you didn't misread AE = 3?

[neon raven]

I checked, it's not wrong

[civic rock]

when you rotate you make a segment that is also equal to 5

[upper echo]

I'm sure. 3+2=5 so, everything is fine

[civic rock]

so the two triangles are congruent

[obsidian harness]

Wait how do you know that the height of D above AB is 5?

[civic rock]

this

[upper echo]

Because the new figure is a deltoid

[maiden brook]

bruh a simple approach is just to find AD and BD why is that not allowed?

[obsidian harness]

Everyone's saying there's a cleverer way than that

[civic rock]

heron's not allowed

that's the challenge

[maiden brook]

why would it not be? but ok

[maiden brook]

ok

[obsidian harness]

I mean, I see by congruence after the rotation that AD = EC

Oh wait no it isn't

Ah I think I'm getting it

That's quite subtle

[maiden brook]

wait dont spoil

wait so why is the height 5..

[maiden brook]

@obsidian harness

or @civic rock

[civic rock]

can you see the congruent triangles?

[maiden brook]

why would they be congruent though

[civic rock]

hmm if I draw it out it'd spoil everything

but since many here has already figured the solution I think I would draw it out anyway

@maiden brook can you see why now?

[maiden brook]

oh yeah that’s pretty nice

[upper karma]

How to solve?

[grave pond]

Note that DG must be parallel to AC.

[upper karma]

Why?

[civic rock]

love how off-scale this is lol

[grave pond]

Each of them intersects CD at the same angle.

[civic rock]

looks like they just drew three equilateral triangles

[upper karma]

Okay so you're saying cd is transversal? And rest I can solve through angles property in transversal

[grave pond]

I'm not completely sure what you mean by "is transversal", but CD is certainly a line that intersects both AC and DG.

[upper karma]

So how to move forward then? After knowing it's parallel?

[civic rock]

you look at AF and its intersections

[grave pond]

Further hint: If I'm right that AC and DG are parallel, ||what does that tell you about ∠FEG?||

[upper karma]

Ok

[upper karma]

Angle Deb will be 2z then angle gef will also be 2z ( vertically opposite). Then I'll apply triangle sum property

[civic rock]

yep you got it