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the ratio of the number of side of 2 regular polygon is 1:2 and the ratio of there interior angle is 3:4 . find the number of side of polygon. |
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Answer» \(\Rightarrow\) The ratio of the sides of two polygon is 1 : 2. \(\Rightarrow\) Let the polygon A have n sides and polygon B have 2n sides. \(\Rightarrow\) The sum of the interior angles of A is (n - 2) x 180º = 180ºn - 360º \(\Rightarrow\)So each interior angle, \(\Rightarrow\)\(\frac{180^on-360^o}{n}\) ............(1) \(\Rightarrow\) Sum of interior angles of B is (2n - 2)×180º = 360ºn - 360º. \(\Rightarrow\) \(\frac{360^on-360^o}{2n}\) ....................(2) \(\Rightarrow\) Now the ratio of the interior angles of A and B. \(\Rightarrow\) \(\frac{180^on-360^o}{n}\): \(\frac{360^on-360^o}{2n}\): : \(\frac{3}{4}\) .................[from (1) and (2)] \(\Rightarrow\) \(\frac{360^on-720^o}{360^on-360^o}\) = \(\frac{3}{4}\) \(\Rightarrow\) \(\frac{360^o(n-2)}{360^o(n-1)}\) = \(\frac{3}{4}\) \(\Rightarrow\) 4n - 8 = 3n - 3 \(\therefore\) n = 5 and 2n = 10 Thus, the number of sides of each polygon is 5 and 10. |
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