What percent of the area of the rhombus ABCD is the area of the circle

Question

GMATH  practice exercise (Quant Class 19)

https://i.postimg.cc/Kv2v8SQL/07-Mar19-7h.gif

What percent of the area of the rhombus ABCD is the area of the circle that is inscribed in it?

(1) Angle BCD is equal to 60 degrees
(2) AB = 4

spandy · Answer

Can someone plz explain how A is the answer?

Posted from my mobile device

chetan2u · Answer

What percent of the area of the rhombus ABCD is the area of the circle that is inscribed in it?

(1) Angle BCD is equal to 60 degrees
Join the diagonals as shown.. Triangle COB is 30-60-90 and sides are in ratio 1: \sqrt{3} :2...
so CO =  a\sqrt{3} , and BO = a, when side BC = 2a.... so AREA of rhombus = 2a*2 a\sqrt{3} 
NEXT take triangle COX, by taking x on tangent .. so again Triangle COX is 30-60-90 and sides are in ratio 1: \sqrt{3} :2...
 Now CO =2*common ratio = a\sqrt{3} , so radius of circle =  \frac{CO}{2}=a\sqrt{3} /2
Thus area of circle =  \pi*(a\sqrt{3}/2)^2=\pi*3a^2/4 ..
Fraction =  \frac{\pi*3a^2}{4*2a*2a*sqrt(3)} ... a will get cancelled out and we can find the fraction
Sufficient

(2) AB = 4
We do not the angles, we cannot say anything..
If all angles are 90, then the fraction will be different and if 60 and 120 then different 
Insufficient

A

fskilnik · Answer

https://i.postimg.cc/RZ2d0s7n/08-Mar19-2x.gif

? = {{{S_{{m{circle}}}}} \over {{S_{{m{rhombus}}}}}}

\left( 1 ight)\,\,\,\left\{ \matrix{
 BCD = {60^ \circ } \hfill \cr 
 ABCD\,\,{m{rhombus}} \hfill \cr} ight.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta \,CBD\,\,{m{and}}\,\,\Delta ABD\,\,{m{equilaterals}}\,\,\,\left( * ight)\,\,\,\,\,

?\,\,\mathop = \limits^{\left( * ight)} \,\,\frac{{{S_{{	ext{circle}}}}}}{{2 \cdot {S_{\Delta {	ext{ABD}}}}}}\,\,\mathop = \limits^{\left( * ight)} \,\,\frac{{\pi \cdot O{E^2}}}{{2 \cdot \left( {\frac{1}{2} \cdot BD \cdot \frac{{BD \cdot \sqrt 3 }}{2}} ight)}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,? = {\left( {\frac{{OE}}{{BD}}} ight)^2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\boxed{\,? = \frac{{OE}}{{BD}}\,}

\left\{ \matrix{
 \,\Delta OEB\,\,\,\,\left  \hfill \cr 
 \,{1 \over 2}BD = OB\,\,\,{m{hyp}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left } \,\,\,\,OE = {{OB \cdot \sqrt 3 } \over 2} \hfill \cr} ight.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {{OE} \over {BD}} = {{OE} \over {2 \cdot OB}} = {1 \over 2}\left( {{{\sqrt 3 } \over 2}} ight)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{m{SUFF}}.

\left( 2 ight)\,\,{m{Insuff}}.\,\,\,\,\left( {{m{geometric}}\,\,{m{bifurcation}}\,{m{,}}\,\,{m{see}}\,\,{m{images}}} ight)

The correct answer is (A).

We follow the notations and rationale taught in the  GMATH  method.

Regards, 
Fabio.

What percent of the area of the rhombus ABCD is the area of the circle

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What percent of the area of the rhombus ABCD is the area of the circle

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