polygons.......

Geometry Level 2

Two regular polygons A B C D ABCD and P T Y F PTYF are such that the ratio of their measures of internal angle is 4 : 3 4:3 and the ratio of their numbers of sides is 2 : 1 2:1 . Find the numbers of sides of the two polygons.

ABCD=7 , PTYF=3.5 ABCD=11 , PTYF=5.5 ABCD=6 , PTYF=3 ABCD=12 , PTYF=6 ABCD=8 , PTYF=4 ABCD=10 , PTYF=5 ABCD=14 , PTYF=7 ABCD=13 , PTYF=6.5

Shruthi Srinivasan by shruthi srinivasan , 13, India

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1 solution

Chris Lewis
Sep 2, 2020

Say the polygons have 2 n 2n and n n sides respectively. The interior angle of a regular n n -gon in radians is given by n − 2 n π \frac{n-2}{n} \pi ; so the equation we want to solve is 3 2 n − 2 2 n π = 4 n − 2 n π 3\frac{2n-2}{2n} \pi = 4\frac{n-2}{n} \pi

Cross-multiplying and tidying, we find the numbers of sides to be n = 5 , 2 n = 10 \boxed{n=5,\;\;2n=10} .

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