Question-155657

Question Number 155657 by mr W last updated on 03/Oct/21

Commented by mr W last updated on 03/Oct/21

Q 154594

Answered by mr W last updated on 03/Oct/21

Commented by mr W last updated on 03/Oct/21

{i t}^{'} s o b v i o u s :

A E = d i a m e t e r o f c i r c l e

Δ A B G \sim Δ F D E \sim Δ F C G

α + β = 45 °

C I = B I = D I = r a d i u s = R

B K = B I \times \cos 2 α = R \cos 2 α

{(\frac{B K}{C I})}^{2} = \cos^{2} 2 α

s i m i l a r l y

{(\frac{D H}{C I})}^{2} = \cos^{2} 2 β = \cos^{2} (90 ° - 2 α) = \sin^{2} 2 α

\Rightarrow {(\frac{B K}{C I})}^{2} + {(\frac{D H}{C I})}^{2} = \cos^{2} 2 α + \sin^{2} 2 α = 1

\frac{Δ_{A B G}}{Δ_{G C F}} = {(\frac{B K}{C I})}^{2}

\frac{Δ_{E D F}}{Δ_{G C F}} = {(\frac{D H}{C I})}^{2}

\frac{Δ_{A B G}}{Δ_{G C F}} + \frac{Δ_{E D F}}{Δ_{G C F}} = {(\frac{B K}{C I})}^{2} + {(\frac{D H}{C I})}^{2} = 1

\Rightarrow Δ_{A B G} + Δ_{E D F} = Δ_{G C F}

\Rightarrow y e l l o w a r e a = h a l f o f c i r c l e

\Rightarrow g r e e n a r e a = h a l f o f c i r c l e

\Rightarrow \frac{y e l l o w}{g r e e n} = 1

Commented by mr W last updated on 03/Oct/21

Commented by amin96 last updated on 03/Oct/21

N i c e s i r r e a l l y g r e a t e

Commented by puissant last updated on 03/Oct/21

W O W M r W i a p p r e c i a t e . .

Commented by Tawa11 last updated on 03/Oct/21

Great sir

Answered by ajfour last updated on 03/Oct/21

Commented by ajfour last updated on 03/Oct/21

(A + D) + E + F = (A + E) + (D + F)

= \frac{r^{2}}{2} (\sin 2 α + \sin 2 β)

= r^{2} \sin θ \cos (α - β) = △_{1}

(B + C) + E + F = (B + E) + (C + F)

= \frac{r^{2}}{2} (\sin (90 ° + 2 α) + \sin (90 ° + 2 β)

= \frac{r^{2}}{2} (\cos 2 α + \cos 2 β)

= r^{2} \cos θ \cos (α - β) = △_{2}

h e n c e f o r θ = 45 °, △_{1} = △_{2}

\Rightarrow A + D = B + C

w e t h u s e x c h a n g e g r e e n A + D

w i t h s a f f r o n B + C

h e n c e

s a f f r o n a r e a = s e m i - c i r c l e a r e a

s a f f r o n a r e a = g r e e n a r e a .

Commented by mr W last updated on 05/Oct/21

n i c e s o l u t i o n!

Facebook Tweet Pin