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Mathematics

Assertion (A) : The number of diagonals of a nonagon = 27.

Reason (R) : The number of diagonals of an n-sided polygon = n(n−1)2−n\dfrac{n(n - 1)}{2} - n.

  1. Both A and R are correct, and R is the correct explanation for A.

  2. Both A and R are correct, and R is not the correct explanation for A.

  3. A is true, but R is false.

  4. A is false, but R is true.

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Answer

The number of diagonals (D) in an n-sided polygon is given by the formula,

D = n(n−3)2\dfrac{n(n - 3)}{2}

For nonagon, n = 9

D = 9(9−3)2=9×62=542\dfrac{9(9 - 3)}{2} = \dfrac{9 \times 6}{2} = \dfrac{54}{2} = 27.

So, assertion (A) is true.

Simplifying the formula for no. of diagonals,

⇒n(n−3)2⇒n(n−1−2)2⇒n(n−1)2−n×22⇒n(n−1)2−n\Rightarrow \dfrac{n(n - 3)}{2}\\[1em] \Rightarrow \dfrac{n(n - 1 - 2)}{2}\\[1em] \Rightarrow \dfrac{n(n - 1)}{2} - \dfrac{n \times 2}{2}\\[1em] \Rightarrow \dfrac{n(n - 1)}{2} - n

So, reason (R) is true.

∴ Both A and R are correct, and R is the correct explanation for A.

Hence, option 1 is the correct option.

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