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
Answer: [spoiler]_____(A)__[/spoiler]
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What percent of the area of the rhombus ABCD is the area of
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fskilnik@GMATH
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GMATH practice exercise (Quant Class 19)
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
Answer: [spoiler]_____(A)__[/spoiler]
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
Answer: [spoiler]_____(A)__[/spoiler]
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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English-speakers :: https://www.gmath.net
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fskilnik@GMATH
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Your Answer
A
B
C
D
E
Global Stats
fskilnik@GMATH wrote:GMATH practice exercise (Quant Class 19)
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
$$? = {{{S_{{\rm{circle}}}}} \over {{S_{{\rm{rhombus}}}}}}$$
$$\left( 1 \right)\,\,\,\left\{ \matrix{
BCD = {60^ \circ } \hfill \cr
ABCD\,\,{\rm{rhombus}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta \,CBD\,\,{\rm{and}}\,\,\Delta ABD\,\,{\rm{equilaterals}}\,\,\,\left( * \right)\,\,\,\,\,$$
$$?\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{{S_{{\text{circle}}}}}}{{2 \cdot {S_{\Delta {\text{ABD}}}}}}\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{\pi \cdot O{E^2}}}{{2 \cdot \left( {\frac{1}{2} \cdot BD \cdot \frac{{BD \cdot \sqrt 3 }}{2}} \right)}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,? = {\left( {\frac{{OE}}{{BD}}} \right)^2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\boxed{\,? = \frac{{OE}}{{BD}}\,}$$
$$\left\{ \matrix{
\,\Delta OEB\,\,\,\,\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right] \hfill \cr
\,{1 \over 2}BD = OB\,\,\,{\rm{hyp}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right]} \,\,\,\,OE = {{OB \cdot \sqrt 3 } \over 2} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {{OE} \over {BD}} = {{OE} \over {2 \cdot OB}} = {1 \over 2}\left( {{{\sqrt 3 } \over 2}} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,{\rm{Insuff}}.\,\,\,\,\left( {{\rm{geometric}}\,\,{\rm{bifurcation}}\,{\rm{,}}\,\,{\rm{see}}\,\,{\rm{images}}} \right)$$
The correct answer is (A).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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