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 $n$ vertices,where $n$ is even,

Vedclass · Accepted Answer

$15$

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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 $n$ vertices,where $n$ is even,the number of rectangles is equal to the number of pairs of diameters. Each diameter connects two opposite vertices. In a $12$-sided regular polygon,there are $n/2 = 12/2 = 6$ diameters. $A$ 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 $^nC_r = \frac{n!}{r!(n-r)!}$. Number of rectangles $= ^6C_2 = \frac{6 	imes 5}{2 	imes 1} = 15$. Thus,there are $15$ such rectangles.

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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