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

[@rishudubey1533]

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

[@PreMath]

You are very welcome! Thanks for your continued love and support! You are awesome, Rishu. Keep smiling👍 Love and prayers from the USA! 😀

[@nurmaryamhibatullah667]

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.

[@venkatk1968]

I could get the step after a few minutes. I never get bored watching these videos

[@RobertHering-tq7bn]

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).

[@soli9mana-soli4953]

Nice solution! Let's promote the non Pythagorean solution

[@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 👍🏻

[@wackojacko3962]

When you grouped 9X as 18X - 9X. Ingenious!!!!

[@DR-kz9li]

Thanks for the lesson. Very clear explanation, as usual. At the end I did: 18-x/x =x/x

[@KAvi_YA666]

Thanks for video.Good luck sir!!!!!!!!!!

[@HappyFamilyOnline]

Very well explained👍 Thanks for sharing😊😊

[@marioalb9726]

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

[@octobrerouge1997]

Merci comme d habitude ❤

[@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.

[@bigm383]

Lovely work Professor!❤🥂👍😊

[@picknikbasket]

Nice, I didn't imagine a quadratic would be involved.

[@alinayfeh4961]

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

[@PreMath]

Thanks for your feedback! Cheers! You are awesome. Keep it up 👍 Love and prayers from the USA! 😀

[@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.

[@harikatragadda]

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

[@vishwassharma3312]

I was also thinking exactly the same.

[@pwmiles56]

With hindsight, you could divide the blue square into 4 equal squares of side 9. If you copy one of these and place it in the position of square ABCD, it's immediately apparent that OD=OE (as they are both diagonals of a rectangle of 2 squares). But I actually solved the quadratic.

[@ybodoN]

First of all, OD=OE as they are both radii of the semicircle...🤔

[@pwmiles56]

@@ybodoN Yes, they have to be, but this shows this is achieved with a square of side 9.

[@jamesrogers4761]

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

[@davidellis1929]

Here's a slight simplification. Let x be the side of the small square, r the radius of the circle. From right triangles with the radius as hypotenuse, we have r^2=9^2+18^2 and r^2=x^2+(x+9)^2, so 2x^2+18x+9^2=18^2+9^2. Cancel out 9^2 from both sides, leaving 2x^2+18x-324=0. Divide by 2 to get x^2+9s-162=0, which factors into (x+18)(x-9)=0. Only x=9 is a positive length, so the green square has area 81.

[@mega_mango]

S = x², side = x Radius= 9√5 because of Pythagorean theorem (9² + 18² = R² = 405). 405 = x² + (9+x)² = 2x² + 18x + 81. x² + 9x - 162 = 0. Positive root of x = 9. x² = 81

[@santiagoarosam430]

Mediante 3 giros de 90 grados aplicados al semicirculo y al cuadrado azul, obtendremos una cruz griega inscrita en el círculo completo, conformada por 12 cuadrados verdes; 2 en cada brazo y 4 en el cuadrado central, cruce de aquellos. Este cuadrado es igual al azul 》Área verde = 324/4 =81 Gracias y un saludo cordial.

[@AmirgabYT2185]

S=81

[@Copernicusfreud]

Yay! I solved it. Area = 81 cm^2.

[@rajbhadursingh8504]

We can also do it by using similarity ..

[@firstteam134]

Just subtract the are of square from the area of the half of the circle, after using pyth. Thm.

[@grandemika]

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

[@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.

[@arthurschwieger82]

Based on this problem, is there, or could you develop, a proof where two squares arranged like these would have an area relationship of 4 to 1? In looking at this, it would seem that no mater the radius of the circle, the relationship would stay the same.

[@EmilMrSweFish]

Yes! Call the area of the big square x^2, and the area of the small square y^2. Then the side lengths are x and y. We want to show that (x^2)/4=y^2, which is the same as showing that x=2y. The radius can be calculated in two ways with the Pythagorean theorem: r^2 = x^2 + (x/2)^2 (equation 1) r^2 = y^2 + (y+x/2)^2 (eq. 2) Set the equations equal to each other x^2 + (x/2)^2 = y^2 + (y+x/2)^2 x^2-y^2 = (y+x/2)^2 - (x/2)^2 Use the formula for difference of squares on both sides. (x+y)(x-y) = (y+x/2+x/2)(y+x/2-x/2) (x+y)(x-y) = (x+y)y x-y = y x = 2y QED

[@marioalb9726]

Esto es un teorema!! SIEMPRE que se tiene esta configuración, el área del cuadrado menor es 1/4 del área del cuadrado central. SIEMPRE !!! Y el lado del cuadrado menor es 1/2 del lado del cuadrado central !!! SIEMPRE. Es un teorema, no se precisan hacer cálculos complicados !!!

[@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. kzbin.info/www/bejne/q3qriKOrapiZp6c

[@himo3485]

Radius is √[9^2+18^2]=√405=9√5 side of Green shaded Square : x x^2+(x+9)^2=405 2x^2+18x+81-405=0 2x^2+18x-324=0 x^2+9x-162=0 (x+18)(x-9)=0 x >0 , x=9 9＊9=81 area of Green shaded Square : 81cm^2

[@alster724]

That was easy

[@giuseppemalaguti435]

l^2+(l+9)^2=18^2+9^2=405..2l^2+18l-324=0..l^2+9l-162=0...l=-9+sqrt729/2=9...A=81

[@philipkudrna5643]

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.

[@neuoylann1909]

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.

[@폴리스다크아미-k5x]

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.

[@zawatsky]

Я сразу заподозрил, что треугольники подобны. Возможно, то же значение 9 можно получить, доказав их подобие.

[@vara1499]

Sir, you have gone about a long way to derive the side of the smaller square to be 9 units. From the figure. BC= 1/2 of BE, which is 18/2 = 9 units. This is a shorter to find the area of the smaller square, which is 9^2= 18sq units. 😊

[@bentels5340]

How do you get that it's half from the picture? Not to scale, remember?

[@vara1499]

@@bentels5340 well, it is very evident. Even he also makes assumptions in many problems that I viewed.

[@bentels5340]

@@vara1499 But what is it evident from? Not to scale means you can't trust what it looks like...

[@vara1499]

@@bentels5340 my assertion gets the same answer/solution. There are videos where the author thinks out of the box to find a solution. If my answer was different, then I am wrong. That's all .

[@adgf1x]

81cm^2

[@kenShuttleworth]

how do you conclude that both (x-9) and (x+18) have to both be equal to 0, as if only one was equal to 0, then multiplied together the result would be 0

[@quigonkenny]

Perhaps he misspoke. What you've determined once you factor it down to (x-9)(x+18)=0 is that either x=9 or x=-18 will get you a result of zero. You can see this by plugging either number back into the original equation or by looking at a graph of the equation, as the graph will cross the x axis at those numbers.

[@Xyz-sd7ub]

81 cm^ 2

[@bentels5340]

Before watching the solution: Area Ab of the blue square = 324 cm² => side S of blue square = 18 cm Radius of semicircle r is from origin to one of the upper corners of the blue square => r² = 18²+9² r² = 405 = 9√5 From center of semicircle to upper left corner of green square is again r Let s = side of green square Then r² = s² + (s+S/2)² r² = s² + (s+9)² 405 = s² + s² 18s + 81 2s² + 18s + - 324 = 0 s² + 9s + - 162 = 0 (s + 18)(s - 9) = 0 s = -18 ∨ s = 9 Since the square is a physical object, its side cannot be negative, so s = 9cm So the area of the green square is 9² cm² = 81 cm² After watching: 👍

[@prossvay8744]

Area green square=9×9=81cm^2

[@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. kzbin.info/www/bejne/q3qriKOrapiZp6c

[@albertofernandez6861]

324cm²/4=81cm² is The area of Green squared.

[@michiel2253]

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.

[@mikefischer8067]

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

[@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

[@paulriley2772]

just divide blue by 4, simples 😉