The number of rectangles that can be obtained | vedclass

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(D) rectangle is formed by two diagonals of a regular polygon that intersect at the center. For a regular polygon with $n = 12$ vertices,the number of

Vedclass · Accepted Answer

$15$

Vedclass · Answer

(D) rectangle is formed by two diagonals of a regular polygon that intersect at the center. For a regular polygon with $n = 12$ vertices,the number of diameters (diagonals passing through the center) is $\frac{n}{2} = \frac{12}{2} = 6$. $A$ rectangle is formed by choosing any $2$ diameters out of these $6$ diameters. The number of ways to choose $2$ diameters from $6$ is given by the combination formula $^nC_r = \binom{n}{r}$. Therefore,the number of rectangles $= \binom{6}{2} = \frac{6 	imes 5}{2 	imes 1} = 15$. Hence,the correct option is $D$.

The number of rectangles that can be obtained by joining four of $12$ vertices of a $12$-sided regular polygon is:

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