Half of a deathly area...

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  • เผยแพร่เมื่อ 20 มี.ค. 2021
  • We look at a nice geometry problem:
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ความคิดเห็น • 1.8K

  • @JohnDoe-jc9zi
    @JohnDoe-jc9zi 3 ปีที่แล้ว +2926

    "AND THATS A GOOD PLACE TO STOP", the fourth brother said calmly.

    • @theandroidguy6032
      @theandroidguy6032 3 ปีที่แล้ว +11

      What a scam wow it just Re direct me to payment and actually i lost my 653$ From my account and the password Given was Wrong wow what a scam salute to u guys.

    • @timoncallens8030
      @timoncallens8030 3 ปีที่แล้ว +51

      @@theandroidguy6032 no way you fell for that

    • @darkstorm1432
      @darkstorm1432 3 ปีที่แล้ว +4

      @@theandroidguy6032 InstaPwn mean instaPayWeNow

    • @Lightwar49
      @Lightwar49 3 ปีที่แล้ว +9

      lmao you guys are dumb enough for falling to that?

    • @rudygabrielperdomo9523
      @rudygabrielperdomo9523 3 ปีที่แล้ว +26

      @@Lightwar49 I think that Makai and Angelo are the same person. Makai makes the claim and Angelo says it works so people will trust Makai. Android Guy must trying to destroy their credibility by saying that it's a scam (it definitely is, I've seen this comment everywhere), even though he probably didn't fall for for it.

  • @elijahzimmerman2053
    @elijahzimmerman2053 2 ปีที่แล้ว +205

    Advanced college geometry and we’re still calling sectors pizza slices. Never change math.

    • @donaldbesong8853
      @donaldbesong8853 ปีที่แล้ว +7

      That has changed. Chapatti is becoming more and more popular in maths.

    • @Ra-vp6fy
      @Ra-vp6fy ปีที่แล้ว

      @@donaldbesong8853 right

    • @ananthpandit2574
      @ananthpandit2574 ปีที่แล้ว +12

      Nahhh tht's high school level

    • @tomaszadamowski
      @tomaszadamowski ปีที่แล้ว +6

      Lol, this is high school level at most

  • @idHawk
    @idHawk 3 ปีที่แล้ว +223

    As a wise man once said: "I wish I were high on potenuse"

    • @hypercodedOld
      @hypercodedOld 3 ปีที่แล้ว +11

      Mr. Jackson, that is enough!

    • @dinupajayaweera287
      @dinupajayaweera287 3 ปีที่แล้ว +5

      Dumbest joke I've ever heard, yet I find myself smiling.

    • @idHawk
      @idHawk 3 ปีที่แล้ว +2

      @@hypercodedOld But, I said it first!

    • @joshhardy5646
      @joshhardy5646 3 ปีที่แล้ว +1

      Gabriel Iglesias approves

    • @aaryanvishwakarma4825
      @aaryanvishwakarma4825 3 ปีที่แล้ว

      Obama wants to know your location

  • @afg99061
    @afg99061 3 ปีที่แล้ว +803

    Me after a long day of studying for finals: "let's get my daily dose of TH-cam to de-stress."
    TH-cam algorithm: here's some math completely unrelated to your degree
    Me: "hmmm yes, I'd like to know what that shaded area is too"

    • @riceagainst1
      @riceagainst1 3 ปีที่แล้ว +13

      I have my second of three 8 hour medical licensing exams in the morning, yet here I am.

    • @hermanplank
      @hermanplank 3 ปีที่แล้ว +5

      @@riceagainst1 a bit late but did you pass the exams? lol

    • @wings4994
      @wings4994 2 ปีที่แล้ว +2

      @@riceagainst1 RESPOND

    • @zayedalmenhali9219
      @zayedalmenhali9219 2 ปีที่แล้ว

      Literally my case 😂

    • @ajitkulkarni1441
      @ajitkulkarni1441 ปีที่แล้ว

      @@riceagainst1 how was it then

  • @ayparillo
    @ayparillo 3 ปีที่แล้ว +1375

    Me who's terrible at math: "Ah yes, of course! Triangles have THREE sides... It's so obvious now!"

    • @favouronwuchekwa
      @favouronwuchekwa 3 ปีที่แล้ว +5

      😂

    • @skrumb
      @skrumb 3 ปีที่แล้ว +6

      ah yes, finally, a cube

    • @Emilia-tan_Manji_Tenshi
      @Emilia-tan_Manji_Tenshi 3 ปีที่แล้ว +2

      what? it isnt?

    • @justanothernick3984
      @justanothernick3984 3 ปีที่แล้ว +6

      This is comedy for me. Me laughing at my own ignorance.
      Is this the start of real life Breaking Bad but instead of chemistry, Walter White does calculus?

    • @BroArmyCommander
      @BroArmyCommander 3 ปีที่แล้ว

      @@justanothernick3984 Did we watch two different versions of Breaking Bad?

  • @TheRealBoroNut
    @TheRealBoroNut 3 ปีที่แล้ว +474

    I managed to follow this perfectly, right up until to the point he said "Hello there...".

    • @moesterer
      @moesterer 3 ปีที่แล้ว +13

      - Did you get that Max?
      - Not quite, Chief.
      - Well, which part didn't you get?
      - The part after you said: now listen carefully.

    • @idHawk
      @idHawk 3 ปีที่แล้ว +10

      General Kenobi

  • @ViralKiller
    @ViralKiller 3 ปีที่แล้ว +540

    now shift the triangles altitude line 1% away from the center of the circle, rotate it clockwise 1 degrees and do it again

    • @AntiFurryNatio
      @AntiFurryNatio 3 ปีที่แล้ว +47

      Great idea!, (*I have no idea what he is telling*)

    • @elrafa964
      @elrafa964 3 ปีที่แล้ว +93

      Some people just like to see the world burn :D

    • @pubsvm7355
      @pubsvm7355 3 ปีที่แล้ว +20

      This is exactly the thing stopping me from studying. They are all scripted problems in school, why should I bother learning them when the actual problems are mostly solved using computer aided stuff these days?

    • @tyleradkins9366
      @tyleradkins9366 3 ปีที่แล้ว +173

      @@pubsvm7355 Without the ability to at least comprehend the principles behind these problems, you can't solve a more complex version. If you're asked to analyze a system that isn't given to you in the form of a problem, you wouldn't even be able to tell enough to get it to where you'd be able to use a computer to solve it. I know this because I'm an engineer and a tutor.

    • @tyleradkins9366
      @tyleradkins9366 3 ปีที่แล้ว +26

      @VK It's entirely possible to solve that problem by hand, you'd just use calculus, not geometry.

  • @rthelionheart
    @rthelionheart 3 ปีที่แล้ว +438

    The fact that the shaded area was found without the aid of integral calculus is a bit surprising.

    • @leandromonteiro8613
      @leandromonteiro8613 3 ปีที่แล้ว +6

      It's implicit

    • @aaaab384
      @aaaab384 3 ปีที่แล้ว +9

      How the fuck is it surprising? There's a triangle and a circle, why on freaking Earth would you use integral calculus, you dummy?!?

    • @rthelionheart
      @rthelionheart 3 ปีที่แล้ว +108

      @@aaaab384 just because you have not the foggiest of ideas how to solve it with integral calculus doesn't mean it cannot be done that way. I am here to learn new things, not to ridicule anyone who uses a different approach than mine.

    • @leandromonteiro8613
      @leandromonteiro8613 3 ปีที่แล้ว +28

      @@aaaab384 with integral calculus it's very much easier to solve this problem

    • @mattweiman5144
      @mattweiman5144 3 ปีที่แล้ว +4

      @@leandromonteiro8613 do go on

  • @zhiar3052
    @zhiar3052 3 ปีที่แล้ว +622

    Before seeing the video, I was planning to do it by calculus:
    1. Set the bottom left point as the origin
    2. Make equation for the circle: (y+r)^2+x^2=r^2
    3. Make equation for the line: y=mx+b, where m=-(3+2.sqrt3)/(2+sqrt3) and b=2+sqrt3.
    4. Find radius by differentiating the circle and setting it equal to to the slope, m.
    5. Set the two equations equal to get the x component of the point of intersection (point of tangency)
    6. Integrate the difference between the equation of the line and the equation of the circle over the range of 0 to the X value that we just found, the result should be the area.

    • @fluffymassacre2918
      @fluffymassacre2918 3 ปีที่แล้ว +28

      You can do that and if you use a computer it is way quicker

    • @pingpongfulldh2308
      @pingpongfulldh2308 3 ปีที่แล้ว +64

      The faster method is by vector calculus using the double integral: int_0^(sqrt(3)/2) int_(sqrt(3)+sqrt(3-x^2))^(-(2+sqrt(3))/(3+2sqrt(3))x+2+sqrt(3)) dy dx and computing it into wolfram alpha

    • @mcbeaulieu
      @mcbeaulieu 3 ปีที่แล้ว +32

      I would have done the same, had I not been in the bathroom when I had the idea 🤣

    • @dimaryk11
      @dimaryk11 3 ปีที่แล้ว +11

      I'm trying integration atm, but it's 5am, and I'm tired lol

    • @SsjRose26
      @SsjRose26 3 ปีที่แล้ว +3

      @@dimaryk11
      I did with simple 10 standard geometry just after waitching thumbnail in 5min

  • @omniyambot9876
    @omniyambot9876 3 ปีที่แล้ว +426

    The feeling of pride solving it and having the same answers but different solutions.

    • @GaryTugan
      @GaryTugan 3 ปีที่แล้ว +7

      ditto :)

    • @tiko952
      @tiko952 3 ปีที่แล้ว +1

      YES

    • @GasNobili
      @GasNobili 3 ปีที่แล้ว +1

      me too, but i only did it in theory, going through the counts I got stuck at first hypotenuse. and still have no idea what he did there lol

    • @omniyambot9876
      @omniyambot9876 3 ปีที่แล้ว

      @@GasNobili we could do it too!!

    • @omniyambot9876
      @omniyambot9876 3 ปีที่แล้ว

      @@FrazerOR yes but using several concepts in solving these kind of problems is fun!

  • @fantiscious
    @fantiscious 2 ปีที่แล้ว +29

    Once you found the third side, I quickly realized that the triangle was a 30-60-90 triangle. 30-60-90 triangles have the property such that the longer leg of the triangle is sqrt(3) times longer than the short leg, and the hypotenuse is 2x as long as the shorter leg.

    • @SyRose901
      @SyRose901 2 ปีที่แล้ว +1

      He used that to find the smaller leg of the triangle that includes the shaded area. Although, yes, he did not realize the pi/6 instantly with that property applied, and instead used the sine theorem.

    • @fantiscious
      @fantiscious 2 ปีที่แล้ว +1

      @@SyRose901 Youre right, i was just hoping to share a tip so everyone can know when to apply that fact. It's good to remember trig values 👍

  • @HighPowerXH
    @HighPowerXH 2 ปีที่แล้ว +50

    I think this solution complicated things a bit. When you find the first Pi/6, you will already know the center angle is Pi/6(similar triangle or simply calculate the shared pi/3 angle), and the edge is 1 by 4+2sqrt(3) - 3+2sqrt(3)

    • @Shadow-Presentations
      @Shadow-Presentations 2 ปีที่แล้ว

      I did that rn on my own too, I only struggled finding radius. Once I saw how to get hypotenuse, yes, its a lot easier our method

  • @kaifengwu6565
    @kaifengwu6565 3 ปีที่แล้ว +127

    The happiness when you figure it out correctly on your own and it turns out to be a simpler method.

    • @ouadii1427
      @ouadii1427 3 ปีที่แล้ว +1

      can you describe the method you used ?

    • @williampeng2962
      @williampeng2962 3 ปีที่แล้ว

      @@ouadii1427 use pythagorean identities and you can see the base triangle is a 30 - 69 - 90 triangle

    • @klaumbazswampdorf1764
      @klaumbazswampdorf1764 3 ปีที่แล้ว +1

      @@williampeng2962 69?

    • @carbon1255
      @carbon1255 3 ปีที่แล้ว +22

      @@klaumbazswampdorf1764 He must be from the sex dimension.

    • @williampeng2962
      @williampeng2962 3 ปีที่แล้ว +1

      @@klaumbazswampdorf1764 60 my bad, typo

  • @davidchung1697
    @davidchung1697 3 ปีที่แล้ว +156

    There is a much easier solution. The key is to see that the triangle formed by connecting the origin of the circle to the point of tangent is similar to the larger triangle. This allows you to solve for R. Also, since the ratio of the legs of the large triangle is SQRT(3) (after rationalizing the denominator), the triangle is a 30-60-90 triangle. You can then use R and the angle to obtain the area.

    • @jayayen3243
      @jayayen3243 3 ปีที่แล้ว +4

      Right, I came here to write this :-)

    • @GreenMeansGOF
      @GreenMeansGOF 3 ปีที่แล้ว

      Nice

    • @subhradipporel285
      @subhradipporel285 3 ปีที่แล้ว

      lmao

    • @tatomar001
      @tatomar001 3 ปีที่แล้ว +16

      @@steveshaff8356 It is a fun trip, also killing a fly with a sledgehammer would be quite spectacular, i'd watch a youtube video about it.

    • @russellharvey7096
      @russellharvey7096 3 ปีที่แล้ว

      @@tatomar001 Someone is working on that right now.

  • @manojmohan9893
    @manojmohan9893 2 ปีที่แล้ว +14

    Hey Michael, I am from India. In my school days I was taught to do a perfect square factorization under a square root sign. It is the same method you used ( taking terms and equating to a and b) but it's done in shorter steps which may be a difficult for beginners or students who lack practice.
    What we do is this -
    2ab√3 = 16√3
    ab =8√3
    a^2 + b^2 = 28
    We got two equations. Now start by selecting values for a and b
    ( a =√3 , b = 8, a^2 + b^2 >28
    a = 2√3 , b = 4, a^2 + b^2 = 28....
    So a=2√3 and b=4

  • @neecow3472
    @neecow3472 3 ปีที่แล้ว +29

    Me: Who has no idea what he's doing
    Also me: Yep, seems about right.

  • @dlevi67
    @dlevi67 3 ปีที่แล้ว +158

    The most colorful Michael Penn video so far.

  • @insouciantFox
    @insouciantFox 3 ปีที่แล้ว +289

    This has to be first use of the half angle formula I’ve ever seen.

    • @anushrao882
      @anushrao882 3 ปีที่แล้ว +1

      Hope it won't be your last :)

    • @hybmnzz2658
      @hybmnzz2658 3 ปีที่แล้ว +22

      More well known is tan(2x) = 2tanx/(1-(tanx)^2)

    • @charlesbromberick4247
      @charlesbromberick4247 3 ปีที่แล้ว +5

      Do you live in a cave?

    • @satvikp3186
      @satvikp3186 3 ปีที่แล้ว +2

      ikr

    • @nikoladjuric9904
      @nikoladjuric9904 3 ปีที่แล้ว +1

      If tan(2x)=p, tan(x)=t
      p=2t/(1-t²), p(1-t²)=2t
      pt²+2t-p=0,
      t=(-1+-_/(1+p²))/p=
      (-1+-1/|cos(2t)|)/(sin(2t)/cos(2t))
      =(-|cos(2t)|+-1)/(sgn(cos(2t))sin(2t))
      If cos(2t)0, it could be
      (-cos(2t)+-1)/(sin(2t))
      Now we check those and see
      (1-cos(2t))/sin(2t)=
      (1-(1-2sin²t))/(2sin(t)cos(t))
      =2sin²(t)/(2sin(t)cos(t)),
      So if sin(t)0 , we can cancel sin(t),
      So that is sin(t)/cos(t)=tan(t)

  • @JLvatron
    @JLvatron 3 ปีที่แล้ว +11

    Great video! I was taken by surprise by your "Good place to stop", cause I thought you would make a common denominator for the solution.

  • @gen3360
    @gen3360 3 ปีที่แล้ว +30

    There is a simpler method on 6:21 , which is by simply splitting 28 such that it forms the addition of two squares i.e. 4 and 2 root 3. I would say it's simpler as it is a direct method and doesn't require many steps. Anyways, nice video.

    • @thepower7803
      @thepower7803 2 ปีที่แล้ว

      how do you think in such a way to spot that? Im convinced it's pretty easy with just a^2 and b^2 but the multiplying by 3 would screw me up... And where does the notation of "root 3" come from anyways? This comment has lead me down quite the math's hole, thankyou.

    • @gen3360
      @gen3360 2 ปีที่แล้ว

      @@thepower7803 oh well the way that you would think like that in this situation is by making the expression within the square root be a copy of the expression a^2+b^2+2ab so that you could simplify the expression and remove the square root, as for the method shown by Mr. Penn as he mentioned it only works in certain situations and not all.

    • @SyRose901
      @SyRose901 2 ปีที่แล้ว +1

      @@thepower7803 One of the most consistent way is to separate the x*sqrt(3) in the initial expression, into multipliers that match the a and b you are trying to find, so 2ab. It's trial and error at that point.

    • @dimosthenisvallis3555
      @dimosthenisvallis3555 ปีที่แล้ว

      Love this. Make it a perect square if u can. Boom!~

  • @ivanrasputin2435
    @ivanrasputin2435 3 ปีที่แล้ว +1322

    Math is beautiful
    I dont even speak english, but I understood everything

    • @jaybeevee6994
      @jaybeevee6994 3 ปีที่แล้ว +228

      says the guy who types english?

    • @dhanvin4444
      @dhanvin4444 3 ปีที่แล้ว +33

      @@jaybeevee6994 good point

    • @ivanrasputin2435
      @ivanrasputin2435 3 ปีที่แล้ว +118

      @@jaybeevee6994 thats a point, então vou falar em português mesmo

    • @darkfire_0579
      @darkfire_0579 3 ปีที่แล้ว +18

      me who speaks English 👁️👄👁️

    • @FTX4816
      @FTX4816 3 ปีที่แล้ว +12

      Dont lie to yourself bro

  • @vilsonandrade8191
    @vilsonandrade8191 3 ปีที่แล้ว +36

    I had concluded my graduation in mechanical engineering 11 years ago, and Google still suggesting mathematics subjects at 1:00Pm.
    - Congratulations by clear explanation 👏🏽👏🏽👏🏽👏🏽👏🏽

  • @dewy367
    @dewy367 3 ปีที่แล้ว +52

    "okay so we are going to this this in a couple of steps"
    Ahh here we go again

  • @nationalstudyacademykim5030
    @nationalstudyacademykim5030 3 ปีที่แล้ว +7

    That was some crazy math! So simple with Trig, algebra, definitions, and geometry, and yet so complicated! Thank you Professor Penn!

  • @igorpereiradasilva4681
    @igorpereiradasilva4681 3 ปีที่แล้ว +55

    I'm physicist and this channel kept my attention with many interesting problems, thank you Michael, great stuff!

  • @josephgranata13
    @josephgranata13 3 ปีที่แล้ว +47

    Just a comment on style - I would’ve noted that the triangle is similar to the standard 1-2-sqrt3 special triangle, done all computations in terms of those simpler quantities, and then multiplied the answer by the factor of 2+sqrt3

    • @GaryTugan
      @GaryTugan 3 ปีที่แล้ว

      my thoughts tooooo

    • @FFF666GP
      @FFF666GP 3 ปีที่แล้ว

      Very astute observation which negates the ‘a+/3b’ manipulation and solving simultaneous non-linear equations.

  • @sakeptik
    @sakeptik 3 ปีที่แล้ว +9

    and then in the corner of the paper, you see:
    *NOT DRAWN ACCURATELY*

  • @basbarbeque6718
    @basbarbeque6718 3 ปีที่แล้ว +50

    It's 3 AM on a sunday.
    I haven't gone to school in 6 years orso.
    Tomorrow I have a D&D session I still need to prep for.
    And before this video I was watching Clips from the Castlevania netflix series.
    My brain: "Mmm indeed, How DOES one calculate that area?"
    TH-cam really is something else.

    • @reinborcheld7210
      @reinborcheld7210 3 ปีที่แล้ว

      Dude...
      Its 2 am on a wednesday night.
      I havent gone to school in 9 years.
      Was watching d&d videos on youtube.
      My brain. Yes! Remember them maths you used to like but never had to use in the past 9 years... Lets get that brain cracking again hahaha!
      I like the parallel of our brain!
      O and how did your session go!?

  • @kay-lideneef28
    @kay-lideneef28 3 ปีที่แล้ว +11

    To find the length 1 at 13:10 you could also use the theorem that tangent lines at a circle are equal. 4+2sqrt3-3-2sqrt3=1

    • @maanavchellani7115
      @maanavchellani7115 3 ปีที่แล้ว +2

      Oh nice! Didn't know that!!!

    • @mr.alhusaini8250
      @mr.alhusaini8250 3 ปีที่แล้ว +2

      It doesn't need all that effort it's a common right angle triangle 📐 that has lengths of ( 1, √3 , 2) or you could say ( k , k√3 , 2k)

    • @emoore06905
      @emoore06905 3 ปีที่แล้ว +1

      He also already established SSS congruence, but the tangent thing works, too

  • @dundarbattaglia4741
    @dundarbattaglia4741 3 ปีที่แล้ว +77

    The main problem is calculate the “radius”
    And the “radius” could be calculated much more easily !!
    Each tangent leg /line to the circle is exactly of same length ( by definition) = 3+2√3
    Since hypotenuse (h) = 4+2√3
    Then, h - 3+2√3 = ( 4+2√3 ) - ( 3+2√3 ) = 1
    *“That 1”* is the length ( of one of the leg) of the small “Top Triangle “ ( *which is congruent with the big triangle* )
    The other Leg is “r “ ( radius)
    So
    1/r = ( 2 + √3 ) / ( 3+2√3 )
    Then r = ( 3+2√3 ) / ( 2 + √3 ) = √3
    =========
    *Note*
    If we see The small “Top Triangle “ is the half of an Equilateral Triangle with each side = 2 and “ height” = √3
    So each angle is 60º
    Then the top angle of the small figure is 60ª ( 1/6 of the 360º)
    Then, everything is very EASY ! and you don't need trigonometry !!
    ===========
    Good moment to remember Einstein : *Imagination is more important than knowledge*
    *KISS principle* ¡!

    • @sumeetsingh2020
      @sumeetsingh2020 3 ปีที่แล้ว

      well done!

    • @maspleben
      @maspleben 3 ปีที่แล้ว

      ... "Corporate want you to find the difference between these two pictures"...

    • @pitchprof65
      @pitchprof65 3 ปีที่แล้ว +1

      Excellent observation!

    • @bernardwodoame9850
      @bernardwodoame9850 3 ปีที่แล้ว +1

      I used this same method

    • @luciangv3252
      @luciangv3252 3 ปีที่แล้ว +2

      Haha he is a beast on math but no in trigonometry

  • @alphapolimeris
    @alphapolimeris 3 ปีที่แล้ว +22

    My reaction seeing the problem:
    "I see no obvious trick.... I am going to parametrize the **** out of this thing !"
    A brutal solution. But hey, it works !

    • @CodAv123
      @CodAv123 3 ปีที่แล้ว +4

      My math teacher always told us that we can use tricks, but before we do so, we have to learn HOW it works and WHY. Only then we can apply formulas to problems that can be solved with one. He actually made very slight changes to problems in tests so they looked like a formula could be applied, but that wasn't really possible. Still some people applied the formula and got the wrong result in turn. I'm quite thankful for his teaching methodology, as after being 20 years out of school now I still remember how to tackle many math problems, even if I don't do it regularly.

  • @ignaciocatalan6592
    @ignaciocatalan6592 3 ปีที่แล้ว +1

    this appeared randomly on my reccomended and it reminded me how much i love geometry and math, this was fun to understand

  • @firefly618
    @firefly618 3 ปีที่แล้ว +18

    I never thought of placing a nested root equal to a + b√n and solving for a and b. Factoring an integer (like -4) and solving for each factor as a system of equations was also a new idea. Quite interesting techniques.

    • @penniesshillings
      @penniesshillings 3 ปีที่แล้ว +1

      That I loved more than the actual problem.

    • @robezoz
      @robezoz 3 ปีที่แล้ว

      It is, I would have tried to complete the square under the square root

    • @albertopalma1663
      @albertopalma1663 3 ปีที่แล้ว

      @@penniesshillings Same here.

    • @therealannakonda
      @therealannakonda 3 ปีที่แล้ว

      I would have used a calculator

  • @ecstasyofgold888
    @ecstasyofgold888 3 ปีที่แล้ว +4

    I love these videos. I love how I remember bits and pieces of algebra and trig from high school along the way and also learn new things I never was exposed to in school. Please keep it up!

  • @Savvy07
    @Savvy07 3 ปีที่แล้ว +4

    This helped in revising my concepts of geometry & Trigonometry.

  • @wongpeter8030
    @wongpeter8030 3 ปีที่แล้ว +1

    Condense way:
    Step 1) You should know the two congruent triangle by tangent property
    Step 2) Get the angle (pi/12) and the hence the radius of circle by trigonometry
    Step 3) Final Area = Big triangle - Area of 2 congruent triangles - Area of half circle + 2 * Sector area with angle(pi/2 - pi/12)

    • @dhunter8286
      @dhunter8286 2 ปีที่แล้ว

      I don't know much math, but from looking at the still shot of the video this is what I hoped he would describe. Thank you

  • @TikeMyson69
    @TikeMyson69 3 ปีที่แล้ว +5

    General solution for triangle with sides a,b,c; side a being the side going through the circle:
    T = ((r^2)/2)((a/b)-arctan(a/b))

  • @wingedshell3518
    @wingedshell3518 3 ปีที่แล้ว +27

    This is the first video I’ve seen of you. And my mind exploded, my eyes widened! I love seeing how everything comes together. I am astonished, this makes me more interested in geometry. I loved this video, it made me really happy! :)

  • @sato88888888
    @sato88888888 3 ปีที่แล้ว +5

    I have to admit, my math education only went up to Calculus 4 and differential equations in college, and after going into medicine none of this makes sense anymore. Definitely a good career move!

  • @mille7476
    @mille7476 2 ปีที่แล้ว +3

    Me: Thinks I'm okay at math and want to proceed with something surrounding math in the future.
    Michael Penn: Crushes my dreams

  • @Qermaq
    @Qermaq 3 ปีที่แล้ว +5

    The first several minutes - I immediately noticed that the long leg is just root3 times the short leg. That made this a lot easier. Also after 10:00 - the circle has two tangents meeting at the pi/6 vertex. They must be collinear. Thus as one is 3 + 2root2, the other must be, so there's an excess on the left of the point of tangency of 1. That's when this got really easy.

  • @blackkk07
    @blackkk07 3 ปีที่แล้ว +53

    To find the radius r and the angle alpha, use the similarity between the left-top triangle and the big green triangle (AA) would be simpler. :)

    • @denismilic1878
      @denismilic1878 3 ปีที่แล้ว +10

      Yes, more than half of the calculations are completely unnecessary, First, calculate angle (2+√3)/(3+2√3)=tan(ϴ)=1/√3 =π/6, Hipotenuse = 2*(2+√3)=(4+2√3). Now you can calculate in your head that one side of the upper triangle is 1 (4+2√3)-(3+2√3), then you can calculate all sides because two angles are the same in similar triangles. Hypotenuse of small triange = 1*2=2 , r =(2+√3)-2 = √3. After that is trivial. Everything without square roots, factorizations, tan half-angle formula, and Pitagoras theorem. Only similar triangles and sin/tan of π/6(30ˇ). To be honest I watch these videos just to see how this guy can make things more complicated than they already are.

    • @davidhoracek6758
      @davidhoracek6758 3 ปีที่แล้ว +8

      You're absolutely right. One thing I learned is that whenever you get a circle in your problem, the first thing you should do is to draw radii to all the interesting points on the circle. Then some relationships just pop out. When I labeled the lengths of the smaller 30-60-90 triangle in terms of r, the radius of the circle, I was able to solve for it without anything fancy like half-angle formulas. Once you know r=√3 you're all set.

    • @Isitar09
      @Isitar09 3 ปีที่แล้ว +5

      For alpha it is even simpler: since we already have theta, 180-90-theta gives the opper angle (clearly 60°). With 180-90-60 gives you alpha (again 30°)

    • @denismilic1878
      @denismilic1878 3 ปีที่แล้ว +4

      @@Isitar09 even simpler this is a problem for gifted 8th graders with no knowledge of trigonometry, you can use the height of a regular triangle for solving it.

    • @koomzog6551
      @koomzog6551 3 ปีที่แล้ว +1

      Exactly what I came here to say. As usual, the solution the teacher gives is unnecessarily complicated. I guess this is because they want to teach advanced formulas, but there are very few examples where they would be useful. I solved this in much simpler way than in the video and reached the same result.

  • @gholmes182
    @gholmes182 3 ปีที่แล้ว

    Loved it. More please. Thanks.

  • @abdullahalaqeel1964
    @abdullahalaqeel1964 3 ปีที่แล้ว

    Thank you very much for demonstrating.

  • @Bodyknock
    @Bodyknock 3 ปีที่แล้ว +6

    13:51 You could find alpha by noticing that alpha plus the other two angles is a straight line and you already know the other two angles are both pi/4 - pi/12 = pi/6. So alpha = pi/2 - pi/3 = pi/6.

  • @ScytheCurie
    @ScytheCurie 3 ปีที่แล้ว +190

    Is NO ONE going to comment on the brilliant Harry Potter reference made in the title?! I guess I have to.

    • @johnjordan3552
      @johnjordan3552 3 ปีที่แล้ว +1

      What exactly is the reference is about?

    • @krettzy3540
      @krettzy3540 3 ปีที่แล้ว +38

      @@johnjordan3552 The deathly hallows is from harry potter, in fact, its the name of the last book and movie(HP and the deathly hallows) It's basically the elder wand(pretty much a very powerful wand) which is the straight line then there is the resurrection stone(the circle) and the invisibility cloak(the triangle) which all come together to form the deathly hallows and its symbol and they play a big part of the final book/2 movies. So, in the video the triangle and all that looks like half of the deathly hallows so it's just referencing that. Hope it's clear. Sorry if I rambled I love HP lol

    • @annad.6382
      @annad.6382 3 ปีที่แล้ว

      @@krettzy3540 OMG thanks ^^

    • @vibhanarayan9668
      @vibhanarayan9668 3 ปีที่แล้ว +1

      Awesome bro I wouldn't have identified it myself awesome 😎 👏 😀 🙌 👌 😄 😎 👏 😀 🙌 👌 😄

    • @GirGir183
      @GirGir183 3 ปีที่แล้ว +1

      The Venn Diagram I'M imagining is the one of A = Those who are intelligent...and B = Those who like the whole harry potter nonsense. I'm guessing A n B is such a tiny sliver of almost-non-existence as to make no difference.

  • @tatoute1
    @tatoute1 3 ปีที่แล้ว +2

    13:50 : one can find triangle area by heron formula and alpha is trivially equal to theta, pi/6 so the arc area can be computed.

  • @krown5666
    @krown5666 3 ปีที่แล้ว +9

    9:47 There is a more easy way to find the radius. The small triangle at the top is also rectangular and the top corner is pi/3. tan(pi/3) = r / (4 + 2 * sqrt(3) - 3 - 2 * sqrt(3)) = r / 1 = r. tan(pi / 3) = sqrt(3) ==> r = sqrt(3). You original way is over engineered.

    • @thelazynarwhal
      @thelazynarwhal 2 ปีที่แล้ว

      What do you mean by the top small triangle is rectangular? I don't understand...

    • @krown5666
      @krown5666 2 ปีที่แล้ว

      @@thelazynarwhal Triangle of the following three dots: the top point, the center of the circle point and the touch point of the circle and the hypotenuse of the big triangle.

  • @wesleydeng71
    @wesleydeng71 3 ปีที่แล้ว +6

    tan(θ) = (2+√3)/(3+2√3) = √3/3 -> θ = pi/6. That saves 5 minutes.😃

  • @conrado1997
    @conrado1997 3 ปีที่แล้ว +5

    5 am, and for some random reason i am watching this...looked so hard in the beggining, thx for making it look easy at the end

  • @antosphere5769
    @antosphere5769 2 ปีที่แล้ว +2

    You can get a solution after finding the radius by drawing a horizontal line from the circle center, which ends up drawing a triangle with two of the desired area plus a quarter of the circle

  • @johnspathonis1078
    @johnspathonis1078 3 ปีที่แล้ว

    Excellent!! Like the pace of the proof.

  • @ThatRedHead717
    @ThatRedHead717 3 ปีที่แล้ว +18

    My man really spent the first 6 minutes to add a one to the bottom leg of that triangle 😂😂😂

    • @pvic6959
      @pvic6959 3 ปีที่แล้ว +7

      but he did it in such a clear way that it made sense to even me. I rather him take the time

    • @draugami
      @draugami 3 ปีที่แล้ว

      If you are watching on a tablet, double tap on the screen to advance 10 seconds. I skipped large portions of this video.

    • @pvic6959
      @pvic6959 3 ปีที่แล้ว +1

      @@draugami if youre on a laptop or computer use the right/left arrow keys. if you do do SHIFT+? you will see the keyboard shortcut menu

  • @junipiter4689
    @junipiter4689 3 ปีที่แล้ว +3

    generalized formula to find the area (with a being the vertical length of the triangle and b being the horizontal length)
    ab/2-(b(b(tan(tan^-1(a/b)))-(((tan^-1(a/b))/360)(b(tan(tan^-1(a/b))2))

  • @gym5959
    @gym5959 2 ปีที่แล้ว

    i enjoyed the problem and your explanation ! great job professor .

  • @johnt.reagan9034
    @johnt.reagan9034 3 ปีที่แล้ว

    That was beautiful! Thanks!

  • @greag1e
    @greag1e 3 ปีที่แล้ว +6

    Went from watching Key and Peele to this, not sure how it happened, but it did.

  • @cuffzter
    @cuffzter 3 ปีที่แล้ว +6

    I think every math teacher should watch this.. just so that they can feel how their students feels after explaining something to them.

    • @tonini617
      @tonini617 3 ปีที่แล้ว +5

      a very clear solution that's well explained?
      Don't project your confusion onto the rest of the world.

    • @ti84satact12
      @ti84satact12 11 หลายเดือนก่อน

      Even though this is a good explanation, I’m a teacher and I understand what you meant!😊

  • @PunmasterSTP
    @PunmasterSTP 3 ปีที่แล้ว

    Another amazing video as always; thank you for sharing!

  • @user-lh1gj6jh7d
    @user-lh1gj6jh7d 3 ปีที่แล้ว

    These videos are so entertaining when you are procrastinating. Also this man have a good voice and detailed instructions, makes people feel calm.

  • @sidPalma
    @sidPalma 3 ปีที่แล้ว +4

    him: find the shaded area
    me: *points at the shaded area* FOUND IT!

  • @jameschannel1451
    @jameschannel1451 3 ปีที่แล้ว +3

    now that im in precalculus, im able to understand this quite well. however, it is still pretty hard and i would hate to have to work this out on my own for an assignment

  • @rockysmith6105
    @rockysmith6105 ปีที่แล้ว

    I saw the thumbnail, and I wondered if we'd be paramaterizing the problem for a general solution amd started w/o watching. I've now watched for some insights bc I knew Michael knew much more what he was doing than I would know myself. The closer here is way better than what I started to do. I was going to compute a general scalene triangle formed by chords from the original right vertex to the tangent of the hypotenuse of the triangle and from said tangent to the diameter of the circle where the circumference crosses the leg it's symmetric about. Which was part one, then add the adjoined chord area at the end of this pseudo sector so ultimately I could subtract the sum from the encompassing triangle which includes the similar right triangle and true sector that I'll use now. Man, that just makes too much sense- thanks for that👍

  • @salihaahmed9500
    @salihaahmed9500 3 ปีที่แล้ว

    That was so fascinating to watch. I’m in awe

  • @fuadjaganjac9193
    @fuadjaganjac9193 3 ปีที่แล้ว +6

    7:33 i saw that peek michael, seems like not even the best of us know the unit circle by heart. don't worry i won't tell anyone ;).

  • @allykid4720
    @allykid4720 3 ปีที่แล้ว +11

    1. From a/b = tan(x) find x = pi/6. Then: c = 2a.
    2. Reflect the triangle on its height to get big triangle c-c-2b with circle inscribed.
    Area of this big triangle:
    A = a×2b/2 = (c+c+2b)× r/2 and get r.
    3. Notice that upper corner has the
    angle pi/3, thus corresponding sector is pi/6.
    4. S = (a - r) × r × sin(pi/6)/2 - pi × r^2/12

  • @jozsefgurzo8777
    @jozsefgurzo8777 2 ปีที่แล้ว

    That was so sweet, thanks!

  • @emulationemperor8924
    @emulationemperor8924 2 ปีที่แล้ว

    A lot of your content I know I could do in high school, or would be able to figure out today if I had some more time. I've done that with a few of your videos that looked interesting or fun to solve at the start. This is the first video that not only would I not have been able to solve this in high school, I'm not even sure I could have solved it with some of my old college knowledge. Always fun to learn how to break down a complicated problem though!

  • @PianoPsych
    @PianoPsych 3 ปีที่แล้ว +9

    You made that much more complicated than necessary.
    Start by examining theta. The tangent of theta is the ratio of our given sides, which easily simplifies to root(3)/3, (multiply the ratio by [(3 - 2*root(3))/(3 - 2*root(3)] )making our triangle similar to a simple right triangle with sides 1, root(3) and 2, in other words, half of the equilateral triangle. That means the hypotenuse of the original triangle is going to have a length that is twice the length of the smaller side or 2*[2 + root(3)] = 4 + 2*root(3), as you discovered with your complicated technique of “un-nesting” the square root.
    After drawing the radius perpendicular to our original triangle and determining the two congruent right triangles created by joining the center of the circle with the 30 degree angle, you could see that the hypotenuse of our original triangle is now divided into two segments, and one of the segments is 3 + 2*root(3), so that means the other segment is
    [4 + 2*root(3)] - [3 + 2*root(3)] = 1. Working with Pi/12 was never necessary. The circular section has an angle of Pi/6 because the small triangle is similar to our original one (they are both right triangles that share the angle at the top). And that quickly reveals the other side and hypotenuse of the small triangle that has side =1 to be root(3) and 2, which shows that the radius of the circle is root(3). We can confirm the hypotenuse of the small triangle is 2 since the original side is
    2 + root(3) and the radius is root(3). No complicated trigonometric formula was required.
    The calculation of the area is then straightforward, just as you describe it.

    • @macacopaco4664
      @macacopaco4664 3 ปีที่แล้ว

      no theta necessary. Only Pythagoras for phuck sake

  • @indrjeetkumar1030
    @indrjeetkumar1030 3 ปีที่แล้ว +11

    Did you know? You are an awesome teacher ❤️

  • @goliathkiller5271
    @goliathkiller5271 3 ปีที่แล้ว

    That was very nice. I really enjoyed it man!
    I'm almost graduating with a masters in Mech Ing, but I haven't done this kind of calculations in a while, and I really miss it!!

  • @wolfram9669
    @wolfram9669 2 ปีที่แล้ว

    Nice geometry problem. I enjoyed this. Thank you.

  • @goodplacetostop2973
    @goodplacetostop2973 3 ปีที่แล้ว +11

    HOMEWORK : This is from the Guts Round of the 4th Annual Harvard-MIT November Tournament.
    Let S be a set of consecutive positive integers such that for any integer n in S, the sum of the digits of n is not a multiple of 11. Determine the largest possible number of elements of S.

    • @nerdiconium1365
      @nerdiconium1365 3 ปีที่แล้ว

      @@zombiekiller7101 I checked with desmos (dear lord, the digit function is a pain in the ass to input, i mean there’s a ceiling of a log on the top of a sigma!!!) and it seems to be true, and it seems to repeat but i’m not sure, no proof here yet

    • @goodplacetostop2973
      @goodplacetostop2973 3 ปีที่แล้ว +1

      @@zombiekiller7101 No

    • @goodplacetostop2973
      @goodplacetostop2973 3 ปีที่แล้ว +3

      SOLUTION
      We claim that the answer is *38*
      This can be achieved by taking the smallest integer in the set to be 999981. Then, our sums of digits of the integers in the set are 45, ... , 53, 45, ... , 54, 1, ... ,10, 2, ... , 10, none of which are divisible by 11.
      Suppose now that we can find a larger set S: then we can then take a 39-element subset of S which has the same property. Note that this implies that there are consecutive integers a−1, a, a+1 for which 10b, ... , 10b+9 are all in S for b=a−1, a, a+1. Now, let 10a have sum of digits N. Then, the sums of digits of 10a+1, 10a+2 , ... , 10a+9 are N+1, N+2, ..., N+9, respectively, and it follows that n≡1 (mod 11).
      If the tens digit of 10a is not 9, note that 10(a+1)+9 has sum of digits N+10, which is divisible by 11, a contradiction. On the other hand, if the tens digit of 10a is 9, the sum of digits of 10(a−1) is N−1, which is also divisible by 11. Thus, S has at most 38 elements.
      Motivation: We want to focus on subsets of S of the form {10a, ..., 10a+ 9}, since the sum of digits goes up by 1 most of the time. If the tens digit of 10a is anything other than 0 or 9, we see that S can at most contain the integers between 10a−8 and 10a+18, inclusive. However, we can attempt to make 10(a−1)+9 have sum of digits congruent to N+9 modulo 11, as to be able to add as many integers to the beginning as possible, which can be achieved by making 10(a−1)+9 end in the appropriate number of nines. We see that we want to take 10(a−1) + 9 = 999999 so that the sum of digits uponadding 1 goes down by 53≡9 (mod 11), giving the example we constructed previously.

    • @user-cr4fc3nj3i
      @user-cr4fc3nj3i 3 ปีที่แล้ว

      May I get some help for this question below?
      If p is prime and n is any natural number.
      Prove that (n+1)^p - n^p - 1 is divisible by p.
      I think that may have something to do with Fermat's little theorem but I don't know what to do with it.

    • @bourhinorc1421
      @bourhinorc1421 3 ปีที่แล้ว

      @@user-cr4fc3nj3i you can see that (something)^p= (something) x ( (something)^[p-1] )
      and using fermats little theorem you have (something)^[p-1] congruent to 1 when p is prime using congruences you can easily do the rest of the problem

  • @kinghassy334
    @kinghassy334 3 ปีที่แล้ว +5

    This is beautiful

  • @csebastian3
    @csebastian3 3 ปีที่แล้ว

    That was a beautiful derivation. Thanks!

  • @calcaware
    @calcaware 3 ปีที่แล้ว +1

    I've always wanted to know the answer to this, but never really tried. Thanks!
    This math genius reminds me of Pauly Shore.

  • @TheDivinepromise
    @TheDivinepromise 3 ปีที่แล้ว +8

    Me as an English major, seeing the first 3 minutes of this…
    Okay I’ve seen too much TH-cam, need to get back reading my books. 😅

    • @ianian4162
      @ianian4162 3 ปีที่แล้ว +2

      Me too, lol. What type of literature are you studying?

    • @PIANOSTYLE100
      @PIANOSTYLE100 3 ปีที่แล้ว

      Me too..I used to love to work calculus problems. I wasn't brilliant, but it was fun..

  • @westerga007
    @westerga007 3 ปีที่แล้ว +3

    Michael, big fan here. But this was needlessly complex. It is easily seen that both tangents on the circle of the yellow triangle are the same length. You even said so in the beginning. Thus the length of that side of the small upper triangle is one. (4+sqrt3)-(3+sqrt3). The vertical side of the small triangle is (2+sqrt3-r). Some simple math shows that r=sqrt3. And then the rest is as you explained.

  • @mrflibble5717
    @mrflibble5717 2 ปีที่แล้ว

    Nice. Thanks Michael.

  • @alcionidas649
    @alcionidas649 2 ปีที่แล้ว

    Excellent!!!! Thank you.

  • @VerSalieri
    @VerSalieri 3 ปีที่แล้ว +16

    The right corner point of the triangle, connected to the center of the circle, forms the bisector of the interior angle (the one on the right obviously) of the triangle.
    I couldn’t ...not say it...sorry.
    But you could have shown it in a much easier way: considering the fact that radii of the same circle are equal, it follows that the center is equidistant from both sides of the angle, thus the center belongs to the bisector of the angle.
    Another way is that the two sides of the angle are tangents to the circle. So the bottom side of the triangle and the segment on the right (formed by the right corner point and the point of tangency) have equal lengths.. because ..that’s what tangents do.. lol (Actually, 2 tangents issued from the same exterior point to the same circle are equal). And the line joining this exterior point and the center of the circle is an axis of symmetry of the figure (just the circle and the 2 tangents with no other elements). So by this symmetry, this axis is a bisector of the angle.
    Also...
    I prefer to “de-nest” the expressions this way:
    Let x=radical(28+16radical3) and y=radical(28-16radical3).
    Then (x+y)^2=x^2+y^2+2xy=64 (just pure calculation)
    and (x-y)^2=x^2+y^2-2xy=48
    which yields that x+y=8 (-8 is rejected since x>y>0) and also x-y=4radical3 (again, -4radical3 is rejected since x>y so x-y>0)
    adding the above 2 equations gives us 2x=8+4radical3 or x=4+2radical3.
    I think using the conjugate is better, leaves less room for guess work or trial and error.
    This gives a really nice idea for a geometry problem for my students. Thanks a lot my friend. Great content.

  • @winky32174
    @winky32174 3 ปีที่แล้ว +3

    Love this one!

  • @aj8__8
    @aj8__8 3 ปีที่แล้ว

    I'm glad this was suggested to me, I finished college two years ago but I missed school, especially math class, this was a fun little problem

  • @orisphera
    @orisphera 3 ปีที่แล้ว +2

    13:40 You can also use similar triangles to figure out the angle

  • @mustafakemalozturk495
    @mustafakemalozturk495 3 ปีที่แล้ว +3

    by the 2nd minute on video i was done applying 30 60 90 triangle and similar triangles he made it far too complicated

  • @swapnamoy6134
    @swapnamoy6134 3 ปีที่แล้ว +44

    Fun fact: you can do this with the help of co-ordinate geometry .

    • @jeevithapatel94
      @jeevithapatel94 3 ปีที่แล้ว

      How so?

    • @swapnamoy6134
      @swapnamoy6134 3 ปีที่แล้ว +10

      @@jeevithapatel94 take the the perpendicular sides as X and y axis and calculate the equation of the hypotenuse line. Take a point on y axis say (0,k) and find the perpendicular distance to the hypotenuse line and Equate with the given condition i.e. both X axis and hypotenuse line are tangent to the circle. You got the radius . Find the area of small triangle containing the required area formed by making a perpendicular to the hypotenuse line from the centre of circle and VOILA! You are more than half done . All that is left is calculate the area of section of circle and substract.

    • @anushrao882
      @anushrao882 3 ปีที่แล้ว

      Nice

    • @vokuheila
      @vokuheila 3 ปีที่แล้ว +5

      To be fair, the vast majority of geometry problems can be bashed with coordinate geometry.

    • @panadrame3928
      @panadrame3928 3 ปีที่แล้ว +2

      @@swapnamoy6134 i don't want to be a fun destroyer but isn't it "Voilà" that you meant instead of wallah ? Or is it some vocabulary I, a simple-minded French people, have not in my bag ?

  • @ahmetatasever8315
    @ahmetatasever8315 3 ปีที่แล้ว

    Thank you very much. Good solution and good presentation.

  • @sdsd2598
    @sdsd2598 3 ปีที่แล้ว

    very clear, thanks a lot

  • @aaaab384
    @aaaab384 3 ปีที่แล้ว +4

    This video should be renamed "How to needlessly overuse trigonometry to solve an elementary geometric problem".

  • @anggalol
    @anggalol 3 ปีที่แล้ว +3

    You upload 1 hour early since yesterday

  • @diegotoledoruy3194
    @diegotoledoruy3194 3 ปีที่แล้ว

    Very nice explanation, very nice video at all. But dont end it like that!

  • @Ploofles
    @Ploofles ปีที่แล้ว

    Very good video, I like your teaching style a lot and I learned a lot from this video. You speak very clearly and explain things well, keep it up.

  • @edwardhaines7917
    @edwardhaines7917 3 ปีที่แล้ว +3

    If processing were interesting, this would be interesting. He glosses over the points of math conceptual connection which turns math into a tic-tac-toe practice.

  • @goodplacetostop2973
    @goodplacetostop2973 3 ปีที่แล้ว +15

    15:42

    • @TrainingCuber
      @TrainingCuber 3 ปีที่แล้ว

      Lol

    • @TrainingCuber
      @TrainingCuber 3 ปีที่แล้ว

      you are really early😂
      I appreciate it

    • @matzew6462
      @matzew6462 3 ปีที่แล้ว

      is that you ? :D

    • @goodplacetostop2973
      @goodplacetostop2973 3 ปีที่แล้ว +1

      @@matzew6462 I'm not Michael Penn. Just a memer that is committed to that joke for too long lol

  • @juliasimoes411
    @juliasimoes411 3 ปีที่แล้ว +1

    this was so interesting, I hadn't tought about separating the angle and creating another triangle. My head just went boooom

  • @HarmeetSingh-jo3sg
    @HarmeetSingh-jo3sg 3 ปีที่แล้ว

    Great video! And so well explained!

  • @khanhnguyengia4168
    @khanhnguyengia4168 3 ปีที่แล้ว +10

    Can u do olympiad geometry problems michael?

    • @trunginhucbao834
      @trunginhucbao834 3 ปีที่แล้ว

      28+ 16 căn 3 =4 (7+4 căn 3)=4 (2^2 + 2.2. căn 3+ 3)=4. (2+ căn 3)^2. Bạn thấy cách này vs cách của Penn cách nào hay hơn nhỉ

  • @digitalconsciousness
    @digitalconsciousness 3 ปีที่แล้ว +4

    Ignore the nit-pickers. It is interesting to see the many ways we could potentially navigate such a problem. Often for mathematicians, knowing the vast array of tools you have at your disposal is the most important thing. Knowing that one thing can be expressed another way, that you can reveal information about A by doing B first, that the more you dissect the given information you have, the more you have to work with, which makes it easier to solve the problem, is ultimately what this channel is about.

  • @soheilaghassemi1172
    @soheilaghassemi1172 ปีที่แล้ว

    Thanks, I appreciate and enjoy.

  • @Roleplay_animal1life
    @Roleplay_animal1life 3 ปีที่แล้ว

    Didn't get the arithmetic part, but just followed through and accepted it as a given :D Then somewhere half way through the video understood solution, no idea how it wasn't that evident from beginning....Thought I should sub, nice channel...apparently I am subbed already :D Thanks for great, professional and fun video!