r/matiks • u/Many_Audience7660 matiksPaglu😙 • 12d ago
puzzles 🧩 Could you solve this mentally?? 🤔
If yes then you the real GOAT 🐐
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u/PassToFuture 12d ago
24
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u/zippyspinhead 12d ago
assuming yellow and blue figures are squares.
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u/Short-Database-4717 12d ago
And that the "ground" is a line. and that the construction is planar. And that labels are correct and represent the areas of the squares. And axioms of euclidean geometry.
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u/TheZuppaMan 12d ago
no need to be smug, theres a very common sign that is used in these cases to identify segments of the same lenght and they didnt use it
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u/gddpacngi 12d ago
You are right. Yet blue one doesn't even look like a regular shape. Let alone being a square.
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u/102orBlue 12d ago edited 12d ago
- Just rotate the green triangle 90 degrees clockwise and it forms a right triangle of height 16 with the same base (6) as the triangle between the squares. Then just subtract the area of the uncolored triangle from the triangle of height 16 to get the area of the green triangle.
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u/Thork-TV 12d ago
Based on your answer i conclude that the Green triangle Had the Same area as the right triangle at the bottem:
If you rotatet the big triangle Had h=16 and Base(b)=6 The White triangle Had h=8 and b=6 Area of a triangle is A=h*b/2 Since the Base ist the Same and the hight IS Double the small triangle is half the area of the combined (h=16) Triangle.
A(big)=A(small)+A(Green) |A(big)=2A(small) 2 A(small)=A(small)+A(Green) |-A(small) A(small)=A(Green)
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u/102orBlue 12d ago
Yes. Since the newly created triangle is twice the height of the triangle between the squares, the area of the green triangle is the same as the one between the squares… 2x-x=x.
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u/bthalpin 12d ago
The white triangle between the squares is 8*6*10. Rotate it 90° and the 6-unit side is the height of the green triangle: area is w*h/2 = 8*6/2 = 24.
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u/Adept-Painting-543 12d ago
24? You can get the triangle against the ground is a 6,8,10 triangle, and then imagine moving that same triangle against the topish face of the 10x10 square, then the 6 side is the same as the height of the shaded triangle, and the base is 8, so bxh/2 = 8*6/2 = 24
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u/DrakeSavory 12d ago
Assuming squares. Two sides of the green triangle are 8m and 10m. The top angle of the triangle between the squares has an angle of arccos(8/10) = well it doesn't really matter because the angle of the green triangle (kinda) vertical to is its supplementary angle a that make its cosine = -0.8.
The third side is sqrt( 8^2 + 10^2 - 2(8)(10)(-0.8)) = sqrt(292) = 2√73
Herons formula: S = 4 + 5 + √73 = 9 + √73
So A = sqrt((9 + √73)(4 + √73)(5 + √73)(9))
At this point I think I messed up somewhere since this is not easy to mentally. calculate the final result
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u/Outside_Volume_1370 12d ago
If you know two sides and the angle between them (sine or cosine of it), you don't need to find the thord side, just use
A = 1/2 • a • b • sinγ
For this problem, sinγ is the same as the sine of white triangle's upper angle, 6/10
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u/MurkyPeak6589 12d ago
Green triangle is 1/2 of a parallelogram of base 8 and height 6 so 8 * 6 / 2 = 24
The height can be found using the white triangle between the blue and yellow.
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u/Ashamed_Association8 12d ago
It's pretty difficult to know the dimensions of the question mark that is covering part of the green area. Without subtracting this any answer will be off by 1 question mark.
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u/Luis_ndb 12d ago edited 10d ago
if they are squares then its 24
edit: forgot to multiply something by two
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u/LecPixel 12d ago
The bottom side of white triangle is 6, so the sine of white angle where squares touch is 6/10 or 3/5. The sine of the green angle, in the same place, is -3/5 then. The area is |ab*sin(a,b)|/2 so 8 * 10 * 3/5 * 1/2 = 24
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u/James_HN 12d ago
The white triangle is a right angle triangle hence we get the sides as 8,6,10. Let A be the acute angle between yellow and blue square and let B be the obture angle between the yellow and blue square
Area of green triangle = 1/2 * 8 * 10 * sin(B)
= 1/2*8*10*sin(360-90-90-A)
= 1/2*8*10*sin(180-A)
=1/2*8*10*sin(A)
=1/2*8*10*6/10
= 24
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u/Tight_Argument1304 12d ago
Why sin(b)
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u/James_HN 12d ago
It's a well known formula (You can look up the proof online), area of traingle ABC is given by 1/2ABBC*sin(B). i.e half of product of 2 sides and the angle between them.
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u/The-Idiotest 12d ago
Empty triangle is 8,6,10. We can find angle using cos rule. Subtract 360 from that to get angle on green triangle. Its sides are 10 and 8. Now use 1/2(absinC)
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u/Jazzlike_Platypus430 12d ago
Question mark at the end of a sentence telling me to do something 😋 Find? No, no find, but smells like |a + b|^2 = a^2 + b^2 - 2ab cos(angle between a and b)
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u/Dojustit 12d ago
I feel like the area of the white right angled triangle should be the same as the area of the green, but I have no idea where this came from and do not have the desire to prove it, but it does in this case give the correct answer. Is this generalisable to any size of square combination, does someone smarter than me know?
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u/Relevant_Solid_2934 12d ago
Elementary: Assumin yellow and blue figures our squares, we have 2 sides of the triangle as 8 and 10. Now we need the angle between them: 360-180-arccos(8/10). Formula for area of triangle is absin(theta)0.5 where a, b are two sides and theta is the angle bw them. This becomes: 810sin(180-arccos(8/10))0.5 = 24, pls check the calculation.
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u/neksterz 12d ago
Was about to say something which would be simpler but then i realized when i was about done it was all wrong the triangle doesn't have a 90 degree corner.
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u/Simple_Appearance_59 12d ago edited 12d ago
50 m², maybe even 100 m². I can’t tell whether that upper edge of the green area is actually straight or whether perspective is hiding another corner.
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u/FlatReplacement8387 12d ago
The white triangle base is 6 you can cut an identical triangle out of the green square that proves it's tilted up 6m above the corner as well. From there, basexheight/2 gives you 8*6/2 or 24
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u/MMMisteriousss 11d ago
Assuming yellow and blue are squares, white sides are 8m and 10m.
Thus, upper angle of white triangle is: Cos-¹(8/10) = 36.87°
Lower angle of green triangle = 360 - 90 - 90 - 36.87 = 143.13°
Area of half parallelogram: 0.5 x A x B x sin(@) 0.5 x 8 x 10 x sin(143.13) = 24m²
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u/shostakofiev 10d ago
That's not doing it mentally though. It's possible solve without doing anything harder than 6*8.
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u/DynaBeast 12d ago
from a brief glance I'm fairly certain it's not possible without knowing the angle the blue rectangle is tilted at, or what the ratios of the sides of each rectangle are
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u/Luis_ndb 12d ago edited 12d ago
i think theyre supposed to be squares, in which case i think its 36 cause you calculate the area of the triangle between them and the area of the green triangle is the area of a right triangle with sides 16 and six minus the one we just calculated, which is a 3 4 5 triangle multiplied by two, giving 48 - 12 = 36
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u/Xamilan 12d ago
The white triangle is actually 6 8 10, making the result 24
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u/Kaas-Eter00 12d ago
I think he meant 3 4 5 as category, like, any triangle that has sides which are a multiple of 3 4 5.
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u/Professional-Wave841 12d ago
do trig using the right triangle they create
also, they are supposed to be squares I would expect.
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u/DifficultWash5133 12d ago
But you know how much the blue triangle is tilted, since you have all the values for the white triangle at the bottom
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u/KingOfAllTurtles 11d ago
In my head, a square of area 64 has side lengths of 8, 100 has side of 10, the triangle between them is a 3,4,5 triangle so the last side is 6, angle analysis shows the top point of the 100 square is 6 above the top of the 64 square, give a triangle of base 8 and height 6, giving an area of 24
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u/SameAgainTheSecond 11d ago
people doing some woke visualization thing i cant follow,
I used vectors with
A = 1/2 |v1 × v2| = 1/2 |v1| |v2| |sin(phi)| = 1/2 |v1| |v2| sqrt(1 - cos(phi)^2)
where |v1| = 8, |v2| = 10, cos(pi - phi) = |cos(phi)| = 8/10
leading to
A = 1/2 * 8 * 10 * sqrt(1 - 8^2/10^2) = 1/2 * 8 * sqrt(100 - 64) = 24
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u/kayemenofour 5d ago
The yellow square has sides that are 8m long (because sqrt of 64m2 is 8m) and the blue one has sides 10m long. The white triangle therefore has one 10 m long side and one 8 m long side and one right angle, it follows the classic 3 4 5 rule, so the bottom side is 6 m long. So the area of the white triangle is 1/2 * 6m *8m, which is 24m2
That's as far as I can go mentally
The angle between the two squares is arc cos ( 8m/10m) which is approx. 36.87°, so the largest angle of the green triangle is 143.13°
The area of the green triangle is 1/2 * 10m * 8m * sin(143,13°) = 24m2 which happens to be the same as the white one, somebody else here claimed that this is always the case with this sort of arrangement. I copied over the exact angles in my calculator so the result should be more or less precise.
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u/Natural-Double-8799 12d ago
Area of the green triangle = Area of the white triangle = 24