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27 Feb 2019, 18:48
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A
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GMATH practice exercise (Quant Class 15)
If the line segment AB is one of the sides of a regular polygon inscribed in the given circle with center O, how many sides does this polygon have?
(1) \(\,\alpha = {30^ \circ }\).
(2) The circumference of the circle is equal to\(\,4\pi \,\).
27-Feb19-8r.gif [ 8.99 KiB | Viewed 1341 times ]
If the line segment AB is one of the sides of a regular polygon inscribed in the given circle with center O, how many sides does this polygon have?
(1) \(\,\alpha = {30^ \circ }\).
(2) The circumference of the circle is equal to\(\,4\pi \,\).
Attachment:
27-Feb19-8r.gif [ 8.99 KiB | Viewed 1341 times ]
28 Feb 2019, 07:22
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fskilnik
\({\rm{regular}}\,\,N{\rm{ - polygon}}\,\,\left( {N \ge 3\,\,{\mathop{\rm int}} } \right)\)
\(? = N\)
\(\left( 1 \right)\,\,\alpha = {30^ \circ }\,\,\,\, \Rightarrow \,\,\,\angle AOB = {60^ \circ }\,\,\,\left( {{\rm{central}}\,\,{\rm{angle}}} \right)\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{regularity}}} \,\,\,\,\,? = {{{{360}^ \circ }} \over {{{60}^ \circ }}} = 6\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)
\(\left( 2 \right)\,\,2\pi r = 4\pi \,\,\, \Rightarrow \,\,\,r = 2\,\,\,:\,\,\left( {{\rm{images}}} \right)\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\alpha = {30^ \circ }\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{regularity}}} \,\,\,\,\,\,? = {{{{360}^ \circ }} \over {{{60}^ \circ }}} = 6\,\,\,\,\left( {{\rm{regular}}\,\,{\rm{hexagon}}} \right) \hfill \cr \\
\,{\rm{Take}}\,\,\alpha = {60^ \circ }\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{regularity}}} \,\,\,\,\,\,? = {{{{360}^ \circ }} \over {{{120}^ \circ }}} = 3\,\,\,\,\left( {{\rm{equilateral}}\,\,{\rm{triangle}}} \right) \hfill \cr} \right.\)
The correct answer is (A).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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