Weekly Problem 44

7138 /www/nrich/html/content/id/7138/ 1 2011-02-01T00:00:01 <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> The diagram shows a regular 9-sided polygon (a nonagon or an enneagon) with two of the sides extended to meet at the point X. What is the size of the acute angle at X? <mdo:image alt="" height="175" src="44%202010.png" width="273"></mdo:image> If you liked this problem, <a href="http://nrich.maths.org/4856&part=">here is an NRICH task</a> which challenges you to use similar mathematical ideas. </mdoxml> <?xml version="1.0" encoding="UTF-8"?> <mdoxml version="1.0"> <mdo:image alt="" height="175" src="44%202010.png" style="text-align: center;" width="273"></mdo:image> The exterior angles of a regular nonagon are $360^{\circ}\div 9 = 40^{\circ}$, whence the interior angles are $180^{\circ} - 40^{\circ}= 140 ^{\circ}$. In the arrowhead quadrilateral whose rightmost vertex is X, three of the angles are $40^{\circ}$, $40^{\circ}$ and $360^{\circ} - 140^{\circ}=220^{\circ}$ and these add up to $300^{\circ}$. So the agle at X is $60^{\circ}$. [It is now posible to see that the entire nonagon can fit neatly inside an equilateral triangle and so the angle X $60^{\circ}$ ] <mdo:image alt="" height="263" src="44%202010%20b.PNG" width="291"></mdo:image></mdoxml> 2 3 0 0 1 0 0 Weekly Problem 44 - 2010 Weekly Problem 44 - 2010 2D Geometry, Shape and Space Angle properties of shapes 2D Geometry, Shape and Space Angles 2D Geometry, Shape and Space Other polygons 2D Geometry, Shape and Space Quadrilaterals Secondary Mapping Document Angles and polygons