Can you solve the inscribed triangle problem?

This is a fun one!
Sources
National Museum of Mathematics tweet
/ 1191411003021504512
Generalization to rectangle Mustafa Kemal Ince tweet
/ 1191433144945135619
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[@gertjan3329]

Great, now engineers will start using the area of that triangle for the gravitational constant. Edit: guys, I get it I mixed up the gravitational constant and average gravitational acceleration on earths surface. I'm sorry, I was very tired when I posted this

[@averagesoul8256]

:))yep i was surprised that i got 9.8 as a result even tho i went with the classic x,l-x aproach and then equations to calculate l But nevertheless since i study engeenring,i therefore am allowed to use the triangle as g:))) Ps:pi squared=g

[@Bunny99s]

Uhm, the gravitational constant is about 6.674 * 10^(-11) m³/(kg*s²) en.wikipedia.org/wiki/Gravitational_constant Too me that looks like you confused "Big G" (the gravitational contstant) with "little g" (the gravitational acceleration at the surface of earth). The 9.81 m/s² is not really a constant. It differs quite a bit even at the surface of earth. Further more it of course decreases the further away from earth you get. The gravitational acceleration can be calculated by g = G * M / r² (where "r" is the distance from the earth center and M is the mass of earth)

[@therandomnomad435]

Acceleration due to gravity, and gravitational constant are two different things.

[@earlmanzor560]

@@Bunny99s trueee. Gravitational constant is different from acceleration due to gravity

[@earlmanzor560]

@@therandomnomad435 truth

[@omerkeskin8559]

Tip: It is just about seeing the sides, and naming them

[@SkiingBiologyGod]

Exactly

[@spiderjerusalem4009]

that's it, i love it when geometry combines with algebra

[@inakibolivar664]

Thanos Becomes Darkseid name one time that geometry didn’t get involved with algebra

[@omerkeskin8559]

@@inakibolivar664 limits

[@inakibolivar664]

Omer Keskin how are limits not related to algebra

[@caiotmz]

This is one of my favorite videos you made. You took a nice problem and solved it in a simple yet satisfying way, showing the general case than applying the numbers. Cheers!

[@mertcanbal1910]

I can't solve any of the questions, but these videos are enjoyable to watch

[@harshitarora2094]

Same lol

[@Mayank-tm2km]

I could solve only one

[@ayeflippum]

*Mertcan Bal* Why are you unable to solve any questions?

[@niluferdincdemirok745]

Dear Presh, thank you for the video.It is highly informative.And its nice to come across to one of my friends Mr Ince as a reference. Have a nice day.

[@Yenmay_24]

My solution also started with denoting the sides and coming up with systems that will end up being quadratic in form. I’m glad I got another ome of these with almost the same approach you did.

[@gofer9156]

There's a simpler and more elegant solution. Note that the rectangle's area (A), is comprised of 2 pieces each of the smaller triangles MINUS the upper-left smaller rectangle (B) formed from the basis of the 3 and 4- triangle. More formally, using Presh's notation: A = 2*(3+4+5)-B. ac/2*bd/2=3*4 -> B:=cd = 48/(a*b) = 48/A So A = 24 - 48/A, with solution A=4(3+sqrt(6)). Subtracting the smaller triangles yields the answer *4*sqrt(6)*

[@kylesheng2365]

Presh was just showing us the more general one, he wasn't only trying to show us the square solution but also the rectangle one

[@gofer9156]

@@kylesheng2365 My solution is general. Just replace 24 with 2(x+y+z) and 48 with 4xy.

[@rahimeozsoy4244]

Where did 48 come from

[@lisew.8474]

@Everybody Nobody Its actually simpler and more intuitive imo. I found the same solution

[@gofer9156]

@Everybody Nobody solving a second-degree equation is straight forward and not covered here. My solution is simpler because it introduces fewer unknowns.

[@briansun6336]

I listed a few equations and used some substitution to solve this. PS: I have seen a question features the same graph but asks a different thing

[@x_abyss]

Hmm, I kinda brute-forced this one. I wrote 5 equations to calculate the length of he side first then subtracted the areas of the triangles. But your approach is more elegant.

[@ahmedshaikha8938]

Post it

[@mauricevassilitch9507]

The 5 equations model gives something quite amusing because at the end, you should have something like (l²-8)(l²-6)=10l² (l => length of the square) But, l² = Area of the square ! So (A(square)-8)(A(square)-6)=10A(square) A(square)=12+4sqroot(6).

[@martinschulz1778]

Yep, I did the same, but I messed up with arithmetic >_< and got the wrong answer :)

[@elonmusk501]

Please post your solution about more than 172 people wants see your idea to solve the problem

[@stuartyeo5354]

Same it took me a page and 10 minutes to do it

[@georgettebeulah4427]

This video make so much sense and meaning and quadratic equation is true I can understand so much from it. Positive thank you. Lv u.

[@darkquantum]

Just have to be careful what areas you label as "x" and "y" since u can rotate the rectangle, esp if its a square. You need x and y to be the areas of the 2 triangles with a side of the rectangle as a leg

[@MattiaElefante]

Good point, you need a way to distinctively identify the quantity 4xy, otherwise the result would change.

[@PedrinhoCruzeirense]

I noticed the same thing, and I was about to write a comment to alert the author of this video. Thumbs up!

[@alexgu177]

he labeled it in the diagram. the reader should be aware of the cyclic asymmetry

[@StevenJacks]

Thank you. The triangles are not all drawn the same. x and y have a whole side length, but z does not. I was wondering what would keep the formula from working with any order of the triangles, and that's why. Thank you. Z is the unique one here.

[@pdaneesh2889]

Thank you for this comment this had helped me...😊

[@gurjot1729]

I want to know how tall is presh TALLwalkar

[@loam]

At least as tall as AT Walkers from Star Wars xD

[@elio7376]

Tall

[@amrelmowaled5786]

I did not laugh 😏

[@ciberiada01]

For me, Presh Tallwalker is nearly as tall as Luke Skywalker.

[@abhijeetm29]

Kyon.. Shadi ke liye rishta bhejna hai kya?

[@lokephoenix1039]

As soon as you said “multiply both sides by ab” I saw exactly where you were going and damn that was beautiful

[@anandarunakumar6819]

Good generalization. For specific case, I used the constraints and set one of short bases as a fraction of length of square. That simplified to a quadratic homogeneous equation of the ratio. Rest of it is algebra, getting the expected result.

[@matthewjames7513]

Hi Presh, I love your videos. I would also love to know how you create them as well! Is it all done in PowerPoint or do you use some other software? Thanks so much :)

[@Ajaysingh56789]

Hey Presh,your teaching style is very good.

[@Porchy.]

my method for intermediate learners: let: A= area of rectangle x= width of rectangle y= height of rectangle a= height of triangle with area=3 b= base of triangle with area=4 ... eq1. ax/2 = 3 hence; ax =6 eq2. by/2 = 4 hence; by =8 eq3. (y-a)(x-b)/2 = 5 hence, xy-by-ax+ab = 10 . take eq.1 and eq.2, multiply to get: abxy = 48 hence ab=48/xy which is now eq.4 . take eq 3. and substitute ax,by, and ab. xy - 8 - 6 + 48/xy = 10. since A = xy , substitute A^2 - 24A + 48 = 0 you get, A= 2.2 and 21.8 we use 21.8 as the logical answer for the area of rectangle . area of middle triangle = A - 3 - 4 -5 which is 9.8 answer is 9.8. .

[@jalalalarabi5307]

Really amazing Mr Presh, my regards sir.

[@barabasbulcsu3102]

You are a true gentleman Mr. Jalal Alarabi!

[@enejidjsi5939]

@@barabasbulcsu3102 No, you both are true gentlemen, Mr. Barabás Bulcsú and Mr. Jalal Alarabi

[@azramaniacNo1]

I love your puzzles. Usually I reach solution in a different way.

[@satyapalsingh4429]

Very good explanation . Thank you, dear professor

[@subswithvideos-io1vx]

"Hello this is pressure cooker"

[@d4django]

I tried for more than hour. Never thought of generalizing. And finding that it was a quadratic equation was genius

[@devondevon4366]

It took me awhile to figure out it was a quadratic equation, A^4 -24x^2 +48x=0

[@jonhreycasingal3200]

I did the long method 😅. didn't know there's a formula that could solve other problems similar to this. thank you! nice explanation!

[@pritamadhikari8203]

I have done it in an another way..Considering each part as different and then adding and substituting the values...It was very awesome to solve...Thank you sir...

[@anandk9220]

I'm happy that I managed to solve this one orally without much effort, although took 15 odd minutes to correct my earlier equation. Honestly it's a very easy problem as other lengths can be easily obtained after assuming square side length as say 'x'. Solving the equation x^4 - 24 (x^2) + 48 = 0, x^2 = 12 + or - 4 √6 Obviously area sum of 3 triangles is 12. So x^2 will be greater than 12. Hence, Required area = 4 √6 square units

[@oorangejuce]

Well..that's one way to start a morning 😄

[@camembertdalembert6323]

for a less complicated quadratic equation : Notice that ab is the area of the rectangle. Therefore ab=x+y+z+w. I also called "t" the quantity x+y+z. If you do this changes in the quadratic equation at 2:58, it becomes less complicated : (t+w)² - 2t(t+w) + 4xy = 0 and then, by using distributivity w² - t² + 4xy = 0 that leads directly to w = sqrt(t²-4xy) = sqrt((x+y+z)²-4x)

[@kilo1265]

I was legit working on this problem for a week banging my head against the wall. Now the solution seems so obvious. At least you got a bunch of other problems for me to try as well.

[@technodrome]

Not gonna lie, that was some dope stuff yo!

[@hemantpatel1413]

That's really an incredible explanation...just wow.

[@lchen1144]

At time about 1:53, one can view the area (ab) of the big rectangle as the sum of three rectangles' areas (2x, 2y, 2z) minus the overlapped rectangle's area [ (2x/b)(2y/a)]. Therefore, ab = (2x+2y+2z)-(2x/b)(2y/a).

[@studyadda4532]

I used to watch your videos and this was the first time I solved the problem by myself very much happy

[@kobeballer]

I had a little different approach. I called AB and BC as x. Then using a system of equations, I expressed FD and ED in terms of x and plugged those into the equation for the area of FDE. This ended up giving me x^4-24x^2+48=0. Solve for x then just calculate the area of the square and subtract the three triangles. Same answer in the end.

[@justinwang3286]

Got it by the same way .

[@hhgygy]

Even easier to solve for x2 instead of x, because that is the area of the entire square. You just don't need x at all.

[@johnnye87]

This is what I was trying to do, but went wrong somewhere. Was a bit annoyed that the video went straight to demonstrating the general case instead of the specific one so I wasn't able to figure out my mistake! Thanks for explaining how you did it, I managed to get there in the end.

[@akshatjhunjhunwala4821]

Love your work sir Love from India

[@ilikeandlovemathsandothers8880]

Thanks from you tube channel I like and love Maths and others.

[@mcgyverlouw3926]

Thank you for the video I just have one question your method seems legit but may we assume that when we transition from a rectangle to a square that the areas of the triangle in the rectangle will reduce by the same factor to acquire the respective areas of the same triangles in the square? I myself don't think so as only the lengths of the top and bottom sides were shorter after the transition. Since the lengths of the left and right sides of the rectangle were not shortened by the same factor as the other two sides,then we really can't expect the areas of the triangle in the rectangle to reduce by the same factor to acquire the areas of the triangles in the square? Thanks so much

[@Noneblue39]

nicely done , i was catching on as he was saying it

[@user-rm3ow6dy7y]

This can be extended to parallelograms with areas S1 S2 S3.

[@egillandersson1780]

I had the particular solution. Thanks for the general one.

[@shantonoyona5687]

Hey i wanna know. Please.

[@ladenhudson2458]

I try to refuse to consider and try the questions you post, but I can not do that. they are so addicting. thank you.

[@PetalGamesStudios]

Got this one! I defined the left side as Y and the top side as X, this gave me 3 formulas using triangles (8 = ay, 6 = bx, 10 = (x-a)*(y-b)). Plugging into the last eqn, I got: 0 = (xy)^2 - 24(xy) + 48. Since xy is the area of the square, I don't need to independently solve X and Y, I instead just solved for the total area which is 12 + 4sqrt(6) and then subtracted the three triangles to get the answer.

[@tera.chaccha2991]

Presh to Albert Einstein : hey Einstein can you figure this out , this problem was given to new borns in China 😂😂 Einstein : 😥😵

[@nikkifraser5487]

The hills are alive with the sound of music doh

[@crazyt1ger08]

People that allow themselves to be ruled by a communist dictatorship that harvests organs from their political opposition and uses terrorism against their own population can't be all that clever......

[@ansalan7734]

@@crazyt1ger08 kzbin.info/www/bejne/ameudJWPmpuae6M

[@crazyt1ger08]

@@ansalan7734 China is certainly into manipulating the figures. Your list is a country's average over the population as a whole, China's 1.4 odd billion population wasn't tested overall, they would have restricted areas of the country to the tests and only allowed the tests to be done to a preselected portion of the country only in order to make it seem China's scores are higher on average than they in fact are in order to 'save face' before the world and to deceive the world into believing they have the best system. A computer glitch here, a few thousand lost test results there. How do I know this? It's what they do with everything.........lie.

[@ansalan7734]

@@crazyt1ger08 kzbin.info/www/bejne/d32YeZ2daJWVla8 So how can you explain it when China emits the most carbon dioxide ?

[@danielreese6185]

That feeling when you approached it differently and get the same answer :)

[@gracyvuna7848]

Beautifull!!! Thanks professor!!!!!!

[@Miscellaneous406]

A really nice problem presh👍

[@taylan2279]

Mustafa Kemal İnce the best mathematician. 👏👏👏

[@user-ud6ud2jl5o]

It's easy to solve,but it's hard to imagine.

[@sonik5436]

exactly 👌✊

[@sanjoychakraborty8638]

Yeah

[@NoovGuyMC]

Yea

[@ojaskumar521]

Yessir

[@enejidjsi5939]

Of course it's easy to solve for you, your name is in chinese 🤣🤣😂

[@user-ly5bc4xd2s]

تمرين جميل. وشرح واضح مرتب . شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا . تحياتنا لكم من غزة فلسطين

[@kazmerpesti]

Cool video! How do you do the letter moving az the end so simple? It is really cool, bit would be pretty hard to do with hand. Is there any program for it?

[@ysgramornorris2452]

Ah yes, of course, quadratic equations! These must've slipped my mind.

[@ultrio325]

Problem:(exists) Mr. Math Man: Δ

[@awesm]

I saw a very similar question with the exact same figure. The side of the square was given 4, and the triangle in the middle was an equilateral triangle. The areas of the smaller triangles were not given.

[@akshaykumarg8703]

Awesome Dude, You did it well

[@rizon8211]

Is there any other way to solve this? Like by finding the side of the square and then . Side squared-(3+4+5)

[@perpetuallyconfused2508]

Rizon yup I did that. got the same answer

[@yili2650]

After watching the whole video 5 times : this question is so easy

[@deonathkumar7387]

This is very simple and easy question, presrawal bhai

[@DoodleNoodle129]

I have no clue why but I love this puzzle so much

[@sracozer2337]

0:38 Mustafa Kemal İnce'ye teşekkürler, izleyen türklerle selam olsun.

[@abdullahyucel553]

Evet duyunca bir farklı hissettirdi hocam :)

[@mustafakemalince1377]

çok teşekkürler

[@xkriolox]

When I found that the square area was 12+- sqrt(6) I thought I was going way off...

[@deerv6913]

I found it in 6 years.

[@ilikeandlovemathsandothers8880]

Thanks from you tube channel. I like and love Maths and others.

[@murdock5537]

Very impressive, many thanks! 😊 Another way solving the problem: divide the rectangle (ab) into four rectangled parts: xy, y(b - x), x(a - y), (a - y)(b - x). xy = ? ax = 8 = xy + x(a - y) by = 6 = xy + y(b - x) (a - y)(b - x) = 10 = ab + xy - 14 = ab +(8/a)(6/b) - 14 → 24 = ab +(48/ab) substitute: k = ab k² - 24k + 48 = 0 k1 = 4(3 + √6) = ab k2= 4(3 - √6) ≠ solution, not big enough - re-substitute: ab = k w = ab - 12 = 4√6 ≈ 9,8 🙂 You can easily see: xy = 24 - ab = 4(3 - √6) Interesting: xy = 12 - w (not obvious) ab = 12 + w (obvious) (w/ab) = (√6 - 2) (w/xy) = (√6 + 2) (ab/xy) = (5 + 2√6)

[@vishalsawarn4234]

Good problem with great solution. Hard to think quadratic in ab...

[@adnanakyuz]

Mustafa Kemal İnce is a math genius 👏👏👏

[@manojjoshi2061]

Sir you are just awesome....... I like the way you teach.......

[@antoniocampos9721]

Very smart solution !

[@NANDHAKUMARNNVK]

Hay I first found the side length of square x(a)= 4.67 by solving polynomial quadratic equation x^4-24x^2+48=0 (actually simple but also got √2.28 as an extra answer)

[@MB-mi6ro]

Именно так сторону квадрата за х, основание треугольника 4 за у, основание треугольника 3 за z, выписать для всех трёх треугольников площадь вот и система, выразить все через х и вот вам уравнение см выше....оно решается и ответ....

[@mrhaci7747]

I found root 12

[@mrhaci7747]

Dont ask how i found it

[@brokenbauccner3945]

Captions : I am Presh to occur 😂😂

[@amansales4226]

The captions always play with his name😂😂

[@zjc7353]

lmao

[@vaibhavgt0]

Hello, this is pressure cooker Lmao

[@user-dailymath100]

너무 좋아하는 채널입니다. 한국에서 수학을 가르치고 있어요. 수학을 잘하기로 유명한 한국학생들도 수학을 어려워 한답니다. 수학이 쉽다고는 할수 있지만 자주 접하면 재미 있어질수 있다고 생각합니다. 화이팅~!!

[@felipeleivax7276]

Seeing the sides is easy to do a 3x3 system and resolve for the sides of the triangles

[@jackylao5560]

I want to ask a geometric problem, which has made me confused for a long time. In a triangle ABC, AB=AC, point D is at AB, then angle ACD=angle BCD, and AD+DC=BC.Find angle ABC My English is not very well, so I don't know if you can understand what I mean. I have solve the problem with sin,cos but I want to know a better solution. I can tell you angle ABC equals 40°. I hope you can see this comment.

[@crystal_royal3405]

Can you mention from where did you get this question??

[@jujuplayboy]

4:52 SUCH A SIMPLE FORMULA ?!? *laughs in equation*

[@senanagayev3623]

Thanks for good solution

[@aram8832]

I am glad my approach was similar to him this time.

[@hubertwilliam6133]

Oh solving the Maths is so amazing and logical 😍

[@goldielocksposition]

Well im impreshed

[@CesarPuerta]

Here's another way to solve: shift the top vertex of y to the top right corner, and the left vertex of x to the bottom left corner. The area under the diagonal is (y+z+xα), where α is the ratio of the base of z and the base of the rectangle. Then α can be derived from (z+xα)/α=y/(1-α).

[@spiderjerusalem4009]

suppose square's lenght = a triangle(area = 4)'s base = b triangle(area=3)'s height = x ½.a.b = 4 ab = 8 b = 8/a ½.a.x = 3 ax = 6 x = 6/a last triangle's base = a-b and its height = a-x its area = 5 ½(a-x)(a-b) = 5 (a-x)(a-b) = 10 a² - ab - ax + bx = 10 a² - (8) - (6) + (8/a)(6/a) = 10 a² - 14 - 10 + 48/a² = 0 a² - 24 + 48/a² = 0 a²(a² - 24 + 48/a²) = a²(0) a⁴ - 24a² + 48 = 0 substitute t = a² a²(a²) - 24a² + 48 = 0 (t)(t) - 24(t) + 48 = 0 t² - 24t + 48 = 0 *here, i'm taking the positive one,* *i mean x1 = (-b + sqrt{b²-4ac})/2a* *, otherwise the square's area would be less than 12* *and if it equaled to less than 12, the unknown area would be negative* *why? Because i substitute* t = a² *which the square's area = a²* t=(-(-24) + sqrt{ (-24)² - 4(1)(48) } )/2(1) = ( 24 + sqrt{ 576-192 } )/2 = (24 + sqrt{384} )/2 = (24 + sqrt{8²×6} )/2 = (24 + 8 sqrt{6} )/2 = ( 2(12 + 4 sqrt(6) )/2 = 12 + 4 sqrt{6} t = a² a² = t = 12 + 4sqrt{6} Unknown area = square's area - 3 - 4 - 5 = a² - 12 = (12 + 4sqrt{6}) - 12 = 4 sqrt{6}

[@leif1075]

And how did you solve the general case?

[@spiderjerusalem4009]

@@leif1075 i beg your pardon?

[@leif1075]

@@spiderjerusalem4009 the case of the rectangle with x y and z areas.

[@haris525]

Lol I solved a lot of these in a 5xxx level university geometry course. We also did Inscribed Circles in triangles. Later I realized I didn’t need to take that class since it was mostly for people who were going to teach math, not applied mathematics. I enjoyed differential geometry much more than geometry.

[@deepTh00ught]

turkler like layinda sayimizi bilelim

[@Tiqerboy]

I liked your comment not because I have any idea what you wrote but because Turkish is cool B-)

[@esrasonmez2691]

Ben hiçbir şey anlamadım

[@crazynurimu9687]

Ehem ehem izninizle, AS BAYRAKLARI AS AS 🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷🇹🇷

[@Monkey_D_Luffyy]

@@Tiqerboy he wrote " Take like Turks and let's know how many we are"

[@kishantalavia6897]

Simply putting heron's formula will also work s=(a+b+c)/2 Area = [s(s-a)(s-b)(s-c)]^0.5 a,b,c are sides of triangle

[@NawazMansuri-qi4ml]

What is the Software name which you used for making animations sir ????

[@priyanusen8215]

Solved it in 1 min

[@ankitasharma4504]

Bahut tej ho maths me

[@leif1075]

How?

[@goldenlesson9374]

first time saw the video is released seconds ago

[@spiderjerusalem4009]

i've seen him video released seconds ago like 2 times

[@brunogreco5084]

We can look for integer triplets x,y,z giving integer w. First solutions by increasing x+y are 2,2,1 ; 2,3,2 ; 2,4,3 ; 2,5,4 ; 3,4,1 ; 3,4,6 ; 2,6,5 ...

[@Unknown-te8fv]

I think you have to get the root of the numbers, then you have individual routes. The triangle is isosceles. You just have to take the mathematical formula for the isosceles triangle, and then you have the area of the isosceles triangle

[@bnmx3456]

vay be türklerden kanalı bilen varmış ismi görünce şaşırdım :D

[@creizi1176]

2:20 you actually factor a -2, but hey, it's still understandable :)

[@Tahgtahv]

Indeed he does. This kind of mistake is often what prompts him to upload corrected videos. It's minor but he strives to be correct when it comes to the math.

[@kasperjoonatan6014]

Not in my opinion. The - is not part of the term. So there's the minus, and then there's 2 times the terms.

[@cezar1948]

@@kasperjoonatan6014 True

[@creizi1176]

@@kasperjoonatan6014 but if you were to factor only a "2", the term would look like +2(-x-y-z)

[@user-pv8mv9st2n]

Just use the cosine rule to find an angle and then for the area apply 1/2×a side×another side×sin(the angle)

[@parthisbest2421]

In this formula , if we solve for general , how would you distinguish between which 2 should be taken as X and y because answer would depend on it as If we change the order , the value of X+y+z remains constant and XY changes so how answer would depend on it .

[@jasper2469]

My classmates be like: 6

[@usingh1]

Writing the answer before watching the solution: Area of triangle BEF (inside the square ABCD) is 9.8

[@yuyo1948]

Genial, ¡un 10, maestro! .

[@rachidballah6113]

Thank you so much i'm from morroco and i love mathematics

[@manuelsanchez-tp8cl]

More examples step to step. So better.

[@protorhinocerator142]

2:26 The area of the rectangle is larger than the area of the triangle, so we keep the positive and discard the negative. Understood, but the negative also solves the equation. What would the triangle look like if we went with the minus sign? Could this even be drawn?

[@botfeeder]

I solved it for the case shown. Then when I started watching his solution and he solved the general case if it's a rectangle. It puzzled me at first, since it seems like there's one more unknown than there are equations. However, a and b are not actually two unknowns, because your final equation to solve contains only the product of a and b. So you wind up with one equation and one unknown, which is a*b. Which means that there is a solution with equal sides in which the sides are length sqrt(a*b), but any rectangle that has that same area a*b will also be a solution for a given x,y,z.

[@tai-shingchau7075]

Let x, a and b be side of the square, bases of right-angled triangle-(4) and right-angled triangle-(3) respectively. My solution with simplified steps: xa = 8, xb = 6 leading to ab = 48/x^2, (x - a)(x - b) = 10; combining the latter two gives x^4 - 24x^2 + 144 = 96 or x^2 - 12 = 4√6, i.e. area of triangle BEF = 4√6.

[@lehtalis]

2:08 , if you still in the square situation (a^2 and not ab) without formula, you can say that a^2 = w + x + y + z = w + 12, so you will get after the distribution and simplication : w - 12 + 48/w + 12 = 0 (w - 12)(w + 12) + 48 = 0 w^2 = 96 w = sqrt(96) = 4sqrt(6)

[@DragonYang01]

Let the side of square to be y and the short side of triangle with area of 4 to be 4x with xy=2. The area to be calculated is y^2-12: We have (y-4x)*(y-3x)=10 => y^2 - 7xy +12x^2=10 => y^4 -24y^2 + 48 =0 => y^4 -24y^2 +144 = 96 Area = y^2-12 = 4 sqrt(6)

[@tonekjansen5597]

The area of the triangle that is to be determined equals sqrt((A+B+C)^2-4AB) with A, B and C the areas of the three given triangles and C the triangle in the bottom-right corner of the rectangle.

[@stephenkormanyos766]

Elegant solution for the general case of a rectangle. There is an easier approach for this example of a square, however. Setting X as the length of one side of the square, using the triangle formula 1/2(b)(h) and the given areas one finds X^4 - 24X^2 = -48 after some basic algebra. Complete the square and find X^2 (the total area of the square) is 12 + 4(6^1/2). Subtract the given areas 3 + 4 + 5 and this also solves w as 4(6^1/2). Thank you for the problem!

[@ledreameryoung9215]

Should not assume it is a square. For versatility, set X for one side of the rectangle and Y for the perpendicular side. If X =Y then it is a square otherwise it is a rectangle. One will realise it does not matter if it is a square or rectangle, the key thing is X.Y = overall area of the square or rectangle and > (3+4+5 = 12).