Geometric Motivations

Geometry Level 4

In the figure shown above, 3 unit circles are squeezed into the smallest square possible.

If the ratio of the area of the yellow region to the area of the white region can be expressed as

a + b + c − 5 π d − π , \dfrac{a + \sqrt b + \sqrt c - 5\pi}{\sqrt d - \pi } ,

where a , b , c , d a,b,c,d are all positive integers with b > c b>c , find a b c d \dfrac{ab}{cd} .

Hint: The first step is to find the square side length.


The answer is 3.

Aziz Alasha by Aziz Alasha , 59, Sudan

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2 solutions


R i g h t v e r t i c a l s i d e o f t h e s q u a r e = r a d i u s o f O 1 + p r o j e c t i o n o f c e n t e r d i s t a n c e O 1 O 2 , 1 5 o t o v e r t i c a l + r a d i u s o f O 2 . W h i t e a r e a = ( a r e a o f e q u i l a t e r a l t r i a n g l e o f c e n t e r s O 1 O 2 O 3 ) − ( s u m o f a r e a s o f ‘ 3 c o l o r e d s e c t o r s o f t h e 3 c i r c l e s ) T h e R a t i o = g r e e n a r e a w h i t e a r e a = ( a r e a o f t h e s q u a r e ) − ( W h i t e a r e a ) − ( s u m o f a r e a s o f t h e t h r e e c i r c l e s . ) w h i t e a r e a = ( 6 + 2 ∗ 2 + 2 6 + 3 ) − ( 3 − π / 2 ) − ( 3 ∗ π / 2 ) 3 − π / 2 = 12 + 32 + 96 + − π − 6 ∗ π 12 − π = a + c + b − 5 ∗ π d − π ∴ a b c d = 12 ∗ 96 12 ∗ 32 = 3. Right~ vertical~ side~ of~ the~ square~ = ~radius~ of~ O_1~ + ~projection~ of~ center~ distance~~O_1O_2,~~ 15^o~~ to~ vertical~ +~ radius~ of~ O_2.\\ White ~area = (area~ of~ equilateral~ triangle~ of~ centers~ O_1O_2O_3) - (sum ~of ~areas~ of`~ 3 ~colored~ sectors~ of~ the~3~ circles)\\ The~ Ratio=\dfrac{green~area}{white~area}\\ =\dfrac{(area~ of~ the~ square) - (White~ area) - (sum~ of~ areas~of~the~three~circles.) }{white~area} \\ =\dfrac{(6+2*\sqrt2+2\sqrt6+\sqrt3) - (\sqrt3-\pi/2) - (3*\pi/2)}{ \sqrt3-\pi/2}\\ =\dfrac{12+\sqrt{32}+\sqrt{96}+-\pi - 6*\pi}{ \sqrt{12}-\pi}\\ =\dfrac{a+\sqrt{c}+\sqrt{b} - 5*\pi}{ \sqrt{d}-\pi}\\ \therefore~\dfrac{ab}{cd}=\dfrac{12*96}{12*32}=\Large \color{#D61F06}{3}.

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Ahmad Saad
May 23, 2017

b=96 (not 95)

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good morning Mr.Ahmad Saad. please recheck your solution, a=12,b=24,c=8,d=12. therefore ab/cd = 24/8=3. your final answer is correct. thankyou v.much

Aziz Alasha - 4 years ago

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Ratio = [12 + 4sqrt6 + 4sqrt2 - 5π]/[2sqrt3 - π] 

      = [12 + sqrt(6×4^2) + sqrt(2×4^2) - 5π]/[sqrt(3×2^2) - π]

      = [12 + sqrt96 + sqrt32 - 5π]/[sqrt12 - π]

     a = 12  ,  b = 96  ,  c = 32  ,  d = 12

I think that you enter the factors of roots to its roote without squaring it.

Ahmad Saad - 4 years ago

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