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by srakute
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this video was sponsored by brilliant let’s say you have a stick that you will cut into three pieces by randomly and independently selecting two points to be locations of the cuts the question is what are the odds the three pieces you create can form a triangle you can

see for example with these three pieces I can create one but if these were the two cuts then it’s impossible to make a triangle now what I found to be an extremely clever solution to this requires one geometry theorem that I bet a lot of people don’t know

if you have an equilateral triangle and pick any point on the interior then draw three line segments from there such that they’re all perpendicular to the three sides the some of those distances will always equal the height of the triangle regardless of where on the interior you pick

here’s why this always works since it’s an equilateral triangle than every side I can call X those line segments will be labeled a B and C and the height of the triangle will be H now the area of the entire triangle would be base or x times height

all over 2 then you’ll notice I can break the bigger triangle up into three smaller triangles using a B and C as the respective heights the area of this bottom one is base x times height a all over to the next triangle would be x times B all

over two and the last one would be X times C all over two and the sum of all of those equals the entire triangle canceling the twos and the X’s we get that a plus B plus C equals H this is known as Vivien ease theorem by the

way but okay how does this help us with our stick problem well first off from geometry you should know that for a triangle to be made the two shortest legs combined in length must be longer than the third if that criteria is met then we can make a

triangle so now we’ll just say that the stick we’re using matches the height of the triangle thus any interior point with those three perpendicular line segments corresponds to cutting that stick in two locations like this interior point corresponds to these two cuts and as we move the interior

point around the overall length stays the same but we change the locations of those two cuts so all the interior points of this triangle correspond to every possible way you could cut this stick in two locations and at this point for example we see it does correspond to

a possible triangle now I’m gonna draw another equilateral triangle in the middle of the bigger one to make a little Triforce here and look at the top triangle for any point inside there the longest line segment or longest piece of the broken stick will definitely be longer than

half the height or total length thus the other two combined will be shorter than that and when the two shorter sides combined don’t exceed the third then you don’t have a possible triangle this applies to anywhere in that upper triangle and by symmetry the same thing can be

said about the other two the middle triangle is the one exception of the longest line segment won’t exceed half the sticks length thus the other two combine must and when the two shorter sides combined exceed the third then you can make a triangle so of the four regions

we see here all equally likely to be the point representation of the two cuts only one of them yields lengths that can make a triangle so the answer to our question is 25% so now this ever comes up in a future interview for you guys hopefully you’ll remember

this video I decided to make this a shorter one but if you like these kinds of puzzles and out-of-the-box thinking then I definitely recommend checking out brilliant org this video sponsor related to this video they actually have an entire series of geometry related courses from beginning to advanced

concepts like that a beautiful geometry course includes fun puzzles like the art gallery problem which is one I really enjoy there’s one in two-dimensional folding or the use of picks theorem which is an application of Euler’s formula and so on or for those who like these advanced puzzles

they have courses like contests math that covers a variety of topics and specific problems which show up in advanced math competitions as everyone says sitting down and challenging yourself with new types of questions is the best way to improve your problem-solving abilities and with all their intuitive animations

along with constant practice problems brilliant org is the perfect resource for doing just that then on top of what we’ve seen they have dozens of other courses of math science and engineering for you to choose from also the first 200 people to sign up with the link below

or by going to brilliant org slash Zack star will get 20% off their annual premium subscription and with that I’m going to end that video there thanks as always to my supporters on patreon social media links to follow me or down below and I’ll see you guys in

the next video

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