Question

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

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

Answer 1

Answer: (A), (B)

The number of isosceles triangles formed by joining the vertices are $196$

$\left(1\right)$ A triangle $A_i,A_j,A_k$ (vertices) is equilateral if $A_i,A_j,A_k$ are equally spaced. Out of $A_1,A_2,.....A_{21}$ we have only $7$ such triplets.
$A_1{A}_8{A}_{15},A_2{A}_9{A}_{16},$.................,$A_7{A}_{14}{A}_{21}.$
Therefore there are only $7$ equilateral triangles.

$\left(2\right)$ Let one of the vertices of the isosceles triangle be $A_1$ now for triangle to be isosceles the other two vertices should be equally spaced from $A_1$
So possibles triplets are $A_1{A}_2{A}_{21},A_1{A}_3{A}_{20},\cdots ,A_1{A}_8{A}_{15},\cdots ,A_1{A}_{11}{A}_{13}$
$\therefore$ number of possible isosceles triangles which are not equilateral and with one vertices as $A_1=10-1=9$
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 $9\times 21=189$.
But the $7$ equilateral triangles are also to be considerd as isosceles.
Hence, total number of isosceles triangle are $196.$