the ratio between the interior and exterior angle of a regular polygon is 3:2, find the the number of sides of a polygon
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the ratio between the interior and exterior angle of a regular polygon is 3:2, find the the number of sides
the ratio between the interior and exterior angle of a regular polygon is 3:2, find the the number of sides of a polygon
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[tex]\huge\rm\underline\pink{GIVEN:}[/tex]
[tex]\huge\rm\underline\pink{TO\:FIND:}[/tex]
[tex]\huge\rm\underline\pink{SOLUTION:}[/tex]
[tex]\large\rm{Let\:the\:Interior\:angle\:be\:3x}[/tex]
[tex]\large\rm{Let\:the\:exterior\:angle\:be\:2x}[/tex]
[tex]\large\rm\bold{\boxed{I\:=\:\frac{(n-2)180}{n}}}[/tex]
[tex]\large\rm{I\rightarrow{Interior\:angle\:of\:regular\:polygon}}[/tex]
[tex]\large\rm{n\rightarrow{Number\:of\:sides\:of\:polygon}}[/tex]
[tex]\large\rm{\implies{\frac{(n-2)180}{n}\:=\:3x}}[/tex]
[tex]\large\rm{\implies{\frac{(n-2)180}{3n}\:=\:x--eq(1)}}[/tex]
[tex]\large\rm\bold{\boxed{E\:=\:\frac{360}{n}}}[/tex]
[tex]\large\rm{E\rightarrow{Exterior\:angle\:of\:Regular\:polygon}}[/tex]
[tex]\large\rm{n\rightarrow{Number\:of\:sides\:of\:polygon}}[/tex]
[tex]\large\rm{\implies{2x\:=\:\frac{360}{n}}}[/tex]
[tex]\large\rm{\implies{x\:=\:\frac{360}{2n}---eq(2)}}[/tex]
[tex]\large\rm{In\:Both\:Equations\:RHS\:Can\:be\:equated}[/tex]
[tex]\large\rm{\implies{\frac{(n-2)180}{3n}\:=\:\frac{360}{2n}}}[/tex]
[tex]\large\rm{\implies{60(n-2)\:=\:\frac{180}{n}}}[/tex]
[tex]\large\rm{\implies{n-2\:=\:\frac{3}{n}}}[/tex]
[tex]\large\rm{\implies{n^2-2n\:=\:3}}[/tex]
[tex]\large\rm{\implies{n^2-2n-3\:=\:0}}[/tex]
[tex]\large\rm{\implies{n^2+n-3n-3\:=\:0}}[/tex]
[tex]\large\rm{\implies{n(n+1)-3(n+1)\:=\:0}}[/tex]
[tex]\large\rm{\implies{(n+1)(n-3)\:=\:0}}[/tex]
[tex]\large\rm{\implies{n\:=\:-1\:(or)\:n\:=\:3}}[/tex]
[tex]\large\rm{n\:=\:-1\:is\:not\:possible\:because\:sides\:of\:Regular\:polygon\:is\:never\:negative}[/tex]
[tex]\large\rm{\therefore{The\:Number\:of\:sides\:=\:3}}[/tex]
Answer:
no of side is 3
Step-by-step explanation:
tq for points