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Question

# Let Ai, i=1,2,3,......21 be the vertices of a 21âˆ’sided regular polygon inscribed in a circle with centre at O. If triangles are formed by joining the vertices of the 21âˆ’sided polygon then

A
The number of equilateral triangles formed by joining the vertices are 7
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B
The number of isosceles triangles formed by joining the vertices are 196
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C
The number of equilateral triangles formed by joining the vertices are 6
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D
The number of isosceles triangles formed by joining the vertices are 186
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Solution

## The correct option is B The number of isosceles triangles formed by joining the vertices are 196(1) A triangle Ai,Aj,Ak (vertices) is equilateral if Ai,Aj,Ak are equally spaced. Out of A1,A2,.....A21 we have only 7 such triplets. A1A8A15, A2A9A16,.................,A7A14A21. Therefore there are only 7 equilateral triangles. (2) Let one of the vertices of the isosceles triangle be A1 now for triangle to be isosceles the other two vertices should be equally spaced from A1 So possibles triplets are A1A2A21,A1A3A20,⋯,A1A8A15,⋯,A1A11A13 ∴ number of possible isosceles triangles which are not equilateral and with one vertices as A1=10−1=9 Similarly for each of the other vertices number of isosceles triangle possible which are not equilateral =9 So, total number of isosceles but non-equilateral triangles are 9×21=189. But the 7 equilateral triangles are also to be considerd as isosceles. Hence, total number of isosceles triangle are 196.

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