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
This topic has expert replies
- fskilnik@GMATH
- GMAT Instructor
- Posts: 1449
- Joined: Sat Oct 09, 2010 2:16 pm
- Thanked: 59 times
- Followed by:33 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
- fskilnik@GMATH
- GMAT Instructor
- Posts: 1449
- Joined: Sat Oct 09, 2010 2:16 pm
- Thanked: 59 times
- Followed by:33 members
Timer
00:00
Your Answer
A
B
C
D
E
Global Stats
$$? = {{{S_{{\rm{circle}}}}} \over {{S_{{\rm{rhombus}}}}}}$$
$$\left( 1 \right)\,\,\,\left\{ \matrix{
BCD = {60^ \circ } \hfill \cr
ABCD\,\,{\rm{rhombus}} \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\Delta \,CBD\,\,{\rm{and}}\,\,\Delta ABD\,\,{\rm{equilaterals}}\,\,\,\left( * \right)\,\,\,\,\,$$
$$?\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{{S_{{\text{circle}}}}}}{{2 \cdot {S_{\Delta {\text{ABD}}}}}}\,\,\mathop = \limits^{\left( * \right)} \,\,\frac{{\pi \cdot O{E^2}}}{{2 \cdot \left( {\frac{1}{2} \cdot BD \cdot \frac{{BD \cdot \sqrt 3 }}{2}} \right)}}\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,? = {\left( {\frac{{OE}}{{BD}}} \right)^2}\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\boxed{\,? = \frac{{OE}}{{BD}}\,}$$
$$\left\{ \matrix{
\,\Delta OEB\,\,\,\,\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right] \hfill \cr
\,{1 \over 2}BD = OB\,\,\,{\rm{hyp}}\,\,\,\,\,\mathop \Rightarrow \limits^{\left[ {{{30}^ \circ },{{60}^ \circ },{{90}^ \circ }} \right]} \,\,\,\,OE = {{OB \cdot \sqrt 3 } \over 2} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = {{OE} \over {BD}} = {{OE} \over {2 \cdot OB}} = {1 \over 2}\left( {{{\sqrt 3 } \over 2}} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.$$
$$\left( 2 \right)\,\,{\rm{Insuff}}.\,\,\,\,\left( {{\rm{geometric}}\,\,{\rm{bifurcation}}\,{\rm{,}}\,\,{\rm{see}}\,\,{\rm{images}}} \right)$$
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)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br