Calculate area of the Green shaded Square | Area of the Blue square is 324 | Fun Olympiad

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Calculate area of the Green shaded Square | Area of the Blue square is 324 | Fun Olympiad

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[rishudubey]

thankyou so much sir ❤
may be you dont know
that your interesting qns make my day
thanks 🙏

[marioalb]

The área is :
Area green = Area blue / 4
Area = 324 / 4
Area = 81cm² ( Solved √ )

Extremely

This is the Theorem of the square inscribed in a semicircle.
It is always so :
Área=1/4 Area
Side=1/2 Side

There 's no need for complicated calculations

[RobertHering-tqbn]

Again this is a very nice puzzle! And, there is another way to find the area via the rule for strings within a circle. To use this we simply complete the half circle downwards to a complete one and add such a small square at the right sight of the bigger square, as well. After determining the side length of the blue square to be 18, using the two strings crossing in C we have (looking for the direction B to E before D over C)...

(18-x)*(18+x) = x*(18+x) <==> 18-x = x <==> 2x = 18 <==> x = 9

And the area of the green shaded square is 81 (cm²).

[harikatragadda]

Applying the Intersecting Chord Theorem at point C,
X*(18+X)=(18+X)*(18-X)
X = 9

[nurmaryamhibatullah]

What a detailed working way! Great job! Because we cannot simply expect the length side of the green square is just a half of the length side of the blue square. It must be proven.

[theoyanto]

I love these quadratic ones mainly because I need the practice. This was another great learning experience for me. The factoring bit at the end I find a bit confusing so usually stick into the quadratic formula calculator app ... Thanks again 👍🏻

[alinayfeh]

Blue shaded square √(324)=18, similar side length of Blue square 18, thereom phythagorus r=√(405), x=9 Green shaded square 81or similar Area two lengths squares 1\4

[TurquoizeGoldscraper]

Instead of numbers, I expressed the size of the green square as a factor of the blue square and determined it to be half the size, or one quarter the area. 324 / 4 = 81.
Also, I noticed you could rotate it 90 degrees and D lines up with F.

[폴리스다크아미-kx]

이번 내용은 볼 것이 없는게... 원 내부에 존재하는 정사각형의 넖이의 합은 원래 반지름의 제곱일 수 밖에 없어요.
어떻게 생겼어도 반지름의 제곱입니다.
차라리 이 이유를 얘기하는게 더 재밌겠네요.

[quigonkenny]

Square BGFE:
A = S²
324 = S²
S = √324 = 18

Triangle ∆OGF:
a² + b² = c²
9² + 18² = r²
81 + 324 = r²
r = √405 = 9√5

Triangle ∆DAO:
a² + b² = c²
s² + (s+9)² = (9√5)²
s² + s² + 18s + 81 = 405
2s² + 18s - 324 = 0
s² + 9s - 162 = 0
s² + 18s - 9s - 162 = 0
s(s+18) - 9(s+18) = 0
(s-9)(s+18) = 0
s - 9 = 0 | s + 18 = 0
s = 9 ✓ | s = -18 ❌ x > 0

Square ABCD:
A = s² = 9² = 81 cm²

[harikatragadda]

A geometric solution can show that the small square side is half the big Square side.
Reflect the semicircle to form a circle and using congruency,
2X= 18
X=9
Image here.

[AnonimityAssured]

Symmetry is our friend here.

Spoiler alert.

It is obvious that if we imagine a point D', such that D'D is a line segment of twice the length of AD, lying parallel to EB, then D'D will have the same length as EF. Meanwhile, it is clear that AO must have the same length as EB. That means that the area of the green square is exactly a quarter of that of the blue square. We don't even need to calculate the square root of 324. We can simply divide it by four.
324 / 4 = 81 square units.

[HappyFamilyOnline]

Very well explained👍
Thanks for sharing😊😊

[grandemika]

We have immediately for simmetry and construction that AD = 1/2 EF
So AD = 18/2 = 9 green area 81.

[mikefischer]

I don’t know why, but after looking it over for about 5 seconds I determined the area was 324/4. Weird!

[michiel]

Sorry I'm a bit of a math nitwit, but given that the green area is a square and adjacent to the blue one, and that D and E are on a circle with centre C, doesn't it simply follow that DC equals EC and as DC also equals CB. And therefore, she's a witch! I mean green is quarter of blue.

[neuoylann]

You can see geometrically from the beginning that the big square is equal to four small squares therefore the area is 324/4 = 81 ! Just inscribe into the circle at the beginning four big squares and then the conclusion is obvious.

[philipkudrna]

The sidelength of the blue square is 18, so half the side is 9. But you don‘t need to know that, the area of the green square must be 1/4th of the blue square (by symmetry) and is therefore 81.

[TheIntellectualRedneck]

I drew this and my instinct was that the solution was that the ratio of the sides of the two squares would be 1/2. The question that comes to mind is about the *next* square (if we insert the largest possible square to the left of the green one), how big will it be? Is there a proof that describes this series of squares of diminishing size?

Edit: I guess that's really a unit circle, the Sine/Cosine laws are in there; that should provide a path.

[jamesrogers]

So is AB always equal to BO? Another amazing circle traight?