Easy to see each of the big triangles represent a quarter. Thus fraction that is green = 1/2 + overlap. The overlapping triangle is similar to big triangles, and w.l.o.g we give the square side length of 2 so we can get the hyp of the big triangles to be sqrt5 and the hyp of the small triangle to be 1. If the side length is scaled down sqrt5 then the area is scaled down sqrt5^2 = 5 so the overlap is 1/5 of the big triangle, or one 20th of the whole square or 5% this leaves 55% that is shaded green. (11/20)
@skwest3 күн бұрын
1) Let the side of the square be 2a. Its area will be 4a². 2) The legs of the two large triangles are 2a and a, making their areas = a² (i.e. 2a•a/2), so, both together would be 2a². 3) They overlap in a small triangle which is similar to the large triangles (left to the reader.) 4) The hypotenuse of the small triangle is a. The hypotenuse of the large triangles is a√5, making the ratio of the sides 1/√5... 5) Therefore the legs of the small triangle are a/√5, and 2a/√5. 6) therefore the area of the overlap is (2a²/5)/2, or a²/5. 7) So the colored area is given as: (area of square) - (sum of areas of large triangle - from #2 above) +(area of overlap - from #6 above). Or: 4a² - 2a² + a²/5 => 2a² + a²/5 => 11a²/5 8) Finally, to get the colored area's percentage of area of the square, we divide the area from #7 by the area of the square, or: (11a²/5)/4a² => 11/20 9) Converting 11/20 to a percentage yields: 55% So, that's the answer: 55% Now to watch the video to see if I got it right. -s.west
@solemnwaltz2 күн бұрын
I did, assuming side lengths of 2 for the square Line 1 is y=x/2+1 Line 2 is y=-2x+4 They intersect at x=(4-1)/(0.5+2) , y=0.5(x)+1 (1.2 , 1.6) The white 5gon and the little triangle at the very top is 1×2÷2=1 area The right hand triangle can be cut into two triangles horizontally From the intersection point earlier, the two new tris are 0.8×0.4÷2 = 0.16 and 0.8×1.6÷2=0.64 4 - 1 - 0.64 - 0.16 = 2.2 2.2/4 is 0.55 So 55%? Or 11/20
@alyssaskier26568 күн бұрын
1. Orange + small blue on the right is a triangle where the base is the right side of the square. So A = b*h/2 = 50% 2. Equation of the line sloping down and to the right is y = .5 - .5 x = (1-x)/2 3. Equation of the perpendicular from the upper right corner to that sloping line is (y - 1) = 2 (x - 1), since it intersects (1,1) 4. y = (1-x)/2 = 2(x-1) + 1, 5. so 1 - x = 4x - 4 + 2; 5x = 3; x = 3/5 6. So the vertex of the right triangle is 3/5 of the way from the left vertex of the orange triangle to the lower right corner of the square. 7. So the area of the triangle id 3/5 * 50% = 30%.
@juanalfaro75229 күн бұрын
tan (BEC) = tan (AED) = 2; let both angles be "a" since they are equal. Then tan (2a) = 2*2/ [1- 2^2] = 4/ [1-4] = -4/3. Now b=BEA = 180-2a -> tan (b) = -(-4/3) = 4/3. If I let all square sides be 4 (so half-side is 2), then BE^2 = 2^2 + 4^2 = 20 -> BE=2*sqrt (5). Now, tan (b) = 4/3 --> sin (b)=4/5 & cos (b) = 3/5. Now EF = BE*cos (b) = 2*sqrt (5) = 6/5 * sqrt (5) and BF = BE*sin (b) = 2*4/5 * sqrt (5) = 8/5 * sqrt (5). Now [BEF] = EF*BF/2 = 6/5 * 8/5 * 5/2 = 24/5. Now [ABCD] = 4^2 = 16 --> BEF % = 24/5/16 = 24/80 = 3/10 = 30%
@carloshenriquedesouzacoelho9 күн бұрын
Isosceles triangle
@coreyyanofsky9 күн бұрын
i fixed the side length at 2 and origin at the point labeled D in the video; then the line segment AE is on the line with equation y = 1 - (x/2) and the line segment BF is on the line with equation y = 2x - 2; they intersect at (6/5, 2/5) and hence AE has length 3/√5, BF has length 4/√5, the triangle has area 6/5, and since in the coordinate system i've chosen the square has area 4 the fraction of the square that is orange is 3/10
@mathu65149 күн бұрын
Alternative solution (no Pythagorean Theorem): kzfaq.info/get/bejne/f5ykfNCHmteZaJs.html Draw a segment from vertex C that intersects segment BE perpendicularly, let's called this intersection point G, then triangles BCG, CEG and BCE are all similar such that the ratio of similarity of triangles CEG and BCG is 1/2, so the ratio of their areas is 1/4. Let x be the area of triangle BCG, then the area of triangle CEG is x/4. Area ▲BCE = Area ▲BCG + Area ▲CEG = x + x/4 = 1/4 → x = 1/5 ▲BCG and ▲ABF are congruent (same angles and same hypotenuse) therefore Area ▲ABF = 1/5 The rest of the solution is the same as in the video
@Bayerwaldler9 күн бұрын
Nice problem! Here my way: Let s be the side length of the square, A it’s area. It‘s easy to show that the upper and lower right triangle each have are A/4. The third right triangle is similar to the other right triangles. It‘s hypotenuse has length s. The other right triangles have hypotenuse length sqrt(1 + (1/2)^2) = s*sqrt(5) / 2. The scaling factor therefore is sqrt(5) / 2. Since the area goes with the square of the scale factor, the area of the third right triangle has only 4/5 the area of the first or second one. Altogether these triangles have an area s^2*(1/4 + 1/4 + 1/5) = 7/10 * s^2. The remaining fourth triangle must therefore have area 3/10 * s^2.
@mathu65149 күн бұрын
nice, that was solution presented in the video but I actually found a way to do the problem without the Pythagorean formula, using only similar triangles and a simple system of equations, I'll post a new video on it soon.
@eddiegaltek15 күн бұрын
Where does the 100x come from? The other numbers - 900, 2,500 & 400 - are fairly obvious, if someone is watching the video they probably don't understand were the 100x came from; if they do they are probably not watching the video.
@mathu651415 күн бұрын
(50 - 𝑥)² = (50 - 𝑥)(50 - 𝑥) = 50*50 - 50𝑥 - 50𝑥 + (-𝑥)(-𝑥) = 2500 - 100𝑥 + 𝑥² (a - b)² = a² - 2ab + b² Thanks, I'll write out more details next time
@MathEducation100M17 күн бұрын
Nice solution
@davidhjedwy17 күн бұрын
if the sides are equal, and the angle is 90 degrees, you can rotate one triangle to get a rectangle
@davidhjedwy17 күн бұрын
implies similarity, then congruence
@xualain312919 күн бұрын
Let the side of the square be a. a*sin(alpha)=3. ….(1) a*sin(belta)=2 or a*cos(alpha)=2. …..(2) Dividing (1) by (2) tan(alpha)=3/2 from which sec(alpha)=sqrt(1+(3/2)^2)=sqrt(13)/2 hence cos(alpha)=2/sqrt(13) From (2) a*2/sqrt(13)=2 then a=sqrt(13) Area=a*a=13
@montyhall-vs3ul10 күн бұрын
wow. why not just realize that the side of the square equals sqrt(2^2 + 3^2). (Pythagorean!) Square that to get the area 13 Btw, the exercise in the vid was to get the area using the most primitive method possible, which doesnt include trig function, or Pythagorean either, for that matter
@youssefelyousfi492923 күн бұрын
(30)u^2
@Antony_V23 күн бұрын
There's different ways to find DC. One of these, for example, is that CH:HD=3:1. Once we have CD=10 shaded area = 1/2*CD*BC = 30 squ
@Antony_V23 күн бұрын
There's a significant mistake, as in the provocative Ramanujan's demonstration (1+2+3+..... infinite = -1/12): integer numbers properties are valid only for a finite value of n.
@plamenpenchev26224 күн бұрын
Upper line segment, x, is 10 (18^2 + 6^2 = x^2) The oblique segment, y, is 6×sqrt(10) (y^2 = 6^2 + 18^2) Sin(t) = 6/(6×sqrt(10)), then 2S = 10×6×sqrt(10)×1/(sqrt(10))
@hongningsuen134824 күн бұрын
There are 3 different methods to find the top side of trapezium x = 10: 1. In triangle ADE, by Pythagoras theorem, x^2 = 6^2 + (18-x)^2 2. Triangle ABC similar to triangle CHD, by corresponding sides proportionality equation x/6sqrt10 = 3sqrt10/18 (DH perpendicular bisector of isosceles triangle ACD) 3. In triangle CHD, x = 3sqrt10/cosB (In triangle ABC, cos B = 18/6sqr10)
@minxythemerciless25 күн бұрын
Easier is to create a rectangle ABCF where F is 6 units above A. Then note that the blue area is (6x18)/2 minus the area of triangle AFD = 6 * (18 - x) / 2 => Area 6x/2 = 30
@davidellis192925 күн бұрын
Once you find that the legs of the shaded triangle have length 10, there's a much simpler way to finish the solution. The shaded area is the trapezoid area less the right triangle at its bottom. This works out to (1/2)(6)(10+18)-(1/2)(6)(18), which is (1/2)(6)(10) = 30.
@mathu651425 күн бұрын
very nice, thank you for watching.
@Paul-sj5dbАй бұрын
I reflected each triangle along their hypotenuse and added two more on the other sides of the square. You end up with a 5*5 square, area 25. If you subtract 4 of the triangles, 4*1/2*2*3 = 12 you get 13.
@mathu6514Ай бұрын
that's a nice way of looking at it, like a flower blooming.
or... you could have half a brain, take one look at the drawing and realize that the two triangles are equal and know that the area of the square is 2 squared plus 3 squared.
@spinbulle5312Ай бұрын
yea even i was thinking the same
@mathu6514Ай бұрын
Brains aren't created with the knowledge of the Pythagorean Theorem already impregnated in them, it had to be learnt by the brain at some point. The point of the video was more to prove that 2 squared plus 3 squared is the area of the square.
@dannypipewrench533Ай бұрын
Do not be mean to the nice math man.
@spinbulle5312Ай бұрын
@@mathu6514 the title of your video reflects to finding the area rather than proving it. If it would have to be done in an Olympiad exam, then this approach shouldn't be applied.
@bpark10001Ай бұрын
You don't need all those constructions! As soon as you determine the congruency of the 2 triangles, you know both are 2 - 3 - √13 triangles. Side of the square is √13, so its area is 13.
@mathu6514Ай бұрын
thank you for the comment, the video wasn't necessarily about finding the area of the square but more to prove the Pythagorean theorem in a roundabout, serendipitous manner. Think of it as an exercise to introduce the Pythagorean theorem to students learning it for the first time.
@northdallashs1Ай бұрын
very obvious
@alessiolaАй бұрын
Very good 👍
@chaosredefined3834Ай бұрын
The area of the triangle is BH/2. We know that the Base is s. The Height of triangle B is half the height of triangle A, by the similar triangle argument you put forward. Combine with the fact that the heights of A and B add up to S, we get that the height of triangle A is 2S/3. So (S)(2S/3)/2 = S^2/3, and therefore, area of triangle / area of square = 1/3.
@pjaj43Ай бұрын
Or a similar argument. By the similar triangles we have established that the rectangle is 2/3 the height of the square and hence 2/3 the area We also know any triangle with one side that of a rectangle and its other corner anywhere along the opposite side is half the area of the rectangle so the area of the red triangle is 2/3 * 1/2 = 1/3 Or to make it difficult, there are two grey overlapping right-angled triangles. One 1/4 the area of the square and the other 1/2, so the bit that isn't the red triangle's has an area area = 1/2+1/4 - the overlap, little similar triangle, (area = 1/2* 1/3 * 1/2 = 1/12) So 1/2 + 1/4 - 1/12 = !/3
@hongningsuen1348Ай бұрын
Join right lower corner to midpoint of top line. A trapezium is formed. The area ratios for the 4 triangles in the trapezium are 1:2:2:4 with total of 9 parts. The area ratio of (square minus trapezium) to (trapezium) is 1:3 hence (square minus trapezium) has 9/3 = 3 parts. Hence fraction of red triangle is 4/(1+2+2+4 +3) = 4/12 = 1/3. The trapezium area ratio can be derived from properties of similar triangles between 2 parallel lines. The general form is T^2:TB:TB:B^2 where T (1 in this problem) is top side ratio and B (2 in this problem) is bottom side ratio, T^2 (1 in this problem) is top triangle area ratio, TB (2 in this problem) is lateral triangles area ratio (equal on both sides), B^2 (4 in this problem) is bottom triangle area ratio. These ratios are very useful in solving area ratio problems.
@PCMlover20236 ай бұрын
I have been reading this book from a few days
@RanjeetKumar-np2pc2 жыл бұрын
How to purchase this book
@sis.lathadavid50112 жыл бұрын
Bring it to me in Tamil
@untermann3223 жыл бұрын
One of my favorite books, thank you!
@PCMlover20236 ай бұрын
I'm also reading it these days
@noam.froehlicher99114 жыл бұрын
Pas mal
@mathu65144 жыл бұрын
merci beaucoup
@celalcesitcioglu54765 жыл бұрын
4
@paras60065 жыл бұрын
How you make your visualization video ?
@mathu65145 жыл бұрын
I'm using Anime Studio to do the animations and Wondershare to edit the videos.
@Jegan_manАй бұрын
@@mathu6514 Hey it's me L. If your reading this, I want to tell you that I am honored to be your student
@mathu6514Ай бұрын
@@Jegan_man thanks L, that really means a lot. You've really grown up and matured a lot these past 4 years, I'm very proud be your teacher. I'm going to do more math livestreams on this channel so stay tune!!!
@Jegan_manАй бұрын
@@mathu6514 Thank you, I look forward to it.
@Jegan_manАй бұрын
@@mathu6514 Hey mr mathu, I added my email address to my channel's about section, I wish to use it to ask you advices in the future. If your reading this, please say hi via email
@ams205185 жыл бұрын
Cool visual demo BUT wish you hadn't shown the answer prior to the demo.